This is almost balanced and ready for playtesting
In balance terms I think you have got this about as balanced as you can without playtesting, except for the 10th level feature. Fixing that then playtesting should be your next step. I expect you will find that some features are overpowered but the lack of slot recovery and the few underpowered feature balance it out overall. As usual I will still go through each feature and give some feedback.
Level 2
Bonus cantrip is good, you are giving access to something druids don't normally get so that makes up for that lack of choice compared to the Circle of the Land. I'm not fussed about the theme of Life/Death from Chaos but that is clearly what you are going for so that's alright. I would grant a more damaging cantrip like firebolt or acid splash to better suit the offensive theme.
1/20 chance of a free Wild Shape is cool and on theme. Though without the forms of Circle of the Moon and no casting in Wild Shape it isn't super useful. This feature is likely slightly underpowered compared to Natural Recovery but not too bad.
Circle Spells
- Crown of Madness: Yes! Totally on theme, a perfect choice
- Shatter: A good AoE attack spell that suits the theme. Good choice.
- Bestow Curse: Pretty good debuff spells that suits the theme. Not bad.
- Vampiric Touch: This is a little more death than Chaos to me. But I get that you want the healing from death type thing. Not a terrible spell choice.
- Death Ward: Not sure about the chaos part of this spell. It feels more like preventing chaos than causing it.
- Greater Invisibility: Not a bad option, it doesn't not suit the theme. Though you do have a lot of concentration spells to compete with it.
- Dawn: Was good before, still good now.
- Destructive Wave: Destructive chaos, basically built of this class. Good choice.
Most of your spell choices are pretty good. Some alternate suggestion for 7th level are: banishment, Evard's black tentacles, confusion, phantasmal killer and vitriolic sphere.
Also the suggestion by Rorp to give chaos bolt is awesome from a theme point of view. Unfortunately it is level 1 and doesn't suit the circle spell progression. I would consider finding a way to fit it in. Maybe instead of the free cantrip at 2nd level you give a 1/rest casting of Chaos Bolt.
Level 6
This is more balanced recharging only on a long rest. I'm not sold on the theme of it but it isn't an issue.
Level 10
Compared to the Circle of the Land this feature is way better. They get immunity to poison and disease, and charm and frightened from fey or elemental. Circle of the Moon get Elemental Wild Shape, Circle of the Shepherd get healing aura for summoned creatures, Circle of Dreams get a limited use teleport. You give advantage on all spell saves, all the time, with no restrictions.
As you say Yuan-ti Pureblood get this as a racial trait from 1st level. While this is support that the feature is not unbalance in general, it is still much stronger than anything the other circles get at this level.
The cantrip as a reaction thing is fine. It might be a little strong if you take my earlier suggestion to give a high damage cantrip but still not a huge deal. Combined with the other benefit is definitely pushes this into overpowered though. I would add some restriction to the magic resistance and cut this part for playtesting.
Level 14
Casting two spells in a turn 1/6 of the time could be really strong. However you will burn spells very quickly. I would limit it to not work on cantrips as that would just make it free damage with no cost. If this proves too strong during playtesting you could consider limiting this action to one weapon attack, Dash, Disengage, Hide, or Use an Object; the same as the action granted by the haste spell.
I love the theme of this ability and think you should stick with it, but be prepared to modify it if it proves too strong.
Why I think spell recovery is important
You say you want this class to be a primary caster druid. Every other primary offensive caster (Circle of the Land druid, wizard, sorcerer, warlock) have a way of regaining some spells on a short rest (sorcerer via sorcery points). Bard and cleric regain other magic-like abilities (Bardic Inspiration and Channel Divinity receptively) on a short rest. Circle of the Moon druids get their main offense (Wild Shape) back on short rests.
Most of these classes also have better offensive cantrips than the druid and don't rely so heavily on spell slots for damage. You are relying on spell slots and not giving a way to recover them. This puts you at a significant disadvantage compared to every other primary caster. That is why I have suggested it multiple times.
There are few reasons the poison damage probably does not apply to the splash damage target
Symbiotic Entity requires you to actually hit your target.
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
The books use the term "hit" not as its standard English meaning but to describe when an attack roll exceeds or equals the AC of a target creature. The secondary effect of green-flame blade does not involve making an attack, and so you have not actually hit this second target.
The secondary damage of green-flame blade likely is not part of the actual melee weapon attack:
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
One way we could hope to know an answer here is by looking at the section on "Critical Hits" which states:
When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together.
We know that critical hits only modify the attack's damage so if a critical hit doubled green-flame blade's secondary damage, that damage would have to be considered part of attack's damage and thus part of the melee weapon attack required by Symbiotic Entity. Unfortunately this is not very clear...
One thing we do know is that damage gated by a saving throw does not double when you score a critical hit. This is supported by this Q/A ("Is Ice Knife's explosion damage affected by rolling a critical?") and also this Q/A ("How does extra damage work for critical hits?"). This is also supported by lead game designer, Jeremy Crawford, though his idea are not official rulings:
Q. DMG Poisons: If you crit with a poison coated weapon, do you double poison dice because crit, or not because saving throw?
A. The intent is no. The saving throw, not the attack, determines whether the poison takes effect after a hit.
However, with green-flame blade there is no save that is gating the additional damage, so perhaps critical hits would double its damage like they do with sneak attack. Crawford has also had input on this specific question stating:
Q. If you critically hit with the attack part of Green Flame Blade, and are greater than level 5, do you roll extra dmg against the 2nd target?
A. The splash damage of green-flame blade isn't affected by the attack critting. Think of the attack as process X & the splash damage as Y.
Thus we at least know the intent is that green-flame blade will not double it's additional die when critting meaning it does not count as part of the attack's damage. Because of this we can conclude that it is not part of the melee weapon attack and thus cannot cause the additional 1d6 damage from Symbiotic Entity.
Symbiotic Entity only effect a target of a melee weapon attack not all of its targets, plural.
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
Because this feature only applies to a singular target which we hit, it cannot apply to more than one person affected by our attack. This idea also would allow us to rule that critically hitting with green-flame blade does not double the additional damage against the second target because critical hits have the same wording, using the singular form of "target".
A note: This does of course require that we assume the phrase "any target" is, in fact, singular. As user @thedarkwanderer has pointed out this phrase could be read as already being plural in which case my argument for point 3 would not apply. There's ambiguity in the meaning as the word "any" in English - It can be used as the plural over the alternative options of "each/every".
A small note on the exceptional state of green-flame blade:
This spell is rather exceptional; you make only one attack but end up affecting multiple creatures. The Sage Advice Compendium has said the following regarding the Ranger's Whirlwind Attack and Volley features:
Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls...
These two features and green-flame blade are the only cases I know of where a single attack affects multiple creatures and so they probably have some odd interactions such as the one asked about in this Q/A ("How does the Sharpshooter feat interact with the Hunter ranger's Volley feature?")
Best Answer
Combat-wise, it feels a bit much.
I recently started a druid. My first intent was to find a way to remain effective in combat despite depleted resources ; I found out that the druid hasn't much to offer, in that regard:
It is indeed very frustrating, and encourages battlefield control over direct damage. Your homebrew subclass fixes all 3 issues, including strong blaster, healer and tank features: it would have been a no-brainer for me, and that's a warning for balance.
Bladesinger comparison
There are not that many full spellcasters with Extra Attack, and Bladesong looks very close to Heart of Fire. It may seem balanced... until you dig a little more:
\$\begin{array}{|c|c|c|} \hline \textbf{Ability} & \textbf{Bladesinger} & \textbf{Circle of flame}\\ \hline \text{max AC using mundane gear} & 23 & \textbf{25} \\ \text{Requirements:} & \textit{mage armor}\text{ spell} & \text{shield, class level 18} \\ & \text{high dexterity & intelligence} & \text{high dexterity or nonmetal half-plate} \\ \hline \text{max AC using magical items} & 25 & \textbf{31} \\ \text{Requirements:} & \textit{robe of the archmagi} & \text{+3 shield & armor, class level 18} \\ & \text{high Dexterity & Intelligence} & \text{high Dexterity or nonmetal half-plate} \\ \hline \text{Hit dice} & \text{d6} & \textbf{d8} \\ \hline \text{Damage control} & \text{reaction & spell slots-based} & \textbf{healing spells} \\ & \text{Damage reduction} & \textbf{reaction-based healing} \\ \hline \text{Concentration}& \text{+Intelligence modifier} & \text{better AC = fewer concentration checks} \\ \hline \end{array} \$
All in all, the Circle of the Flame druid vastly outperforms its cousin - and can even do so with little ASI investment, should you find a proper armor.
Circle of Land & Circle of Spores comparisons
The Circle of Spores, from Unearthed Arcana, is probably an attempt to offer wannabe-melee druids an alternate to the circle of the moon's combat forms. Let's include it here:
\$\begin{array}{|c|c|c|} \hline \textbf{Subclass features} & \textbf{Circle of the Flame} & \textbf{Circle of the Land} & \textbf{Circle of Spores} \\ \hline \text{level 2} & \textbf{$fire\ bolt$} & \text{one druid cantrip} & \textit{chill touch} \\ & \textbf{1-minute huge melee buff} & \text{Natural Recovery} & \text{10-min small melee buff} \\ & & & \text{reaction-based poison damage} \\ \hline \text{level 6} & \textbf{Extra Attack} & \text{ignoring difficult terrain} & \text{1-hp zombies} \\ \hline \text{level 10} & \textbf{+Wis mod to fire, concentration spells} & \text{various immunities} & \text{AoE spores} \\ & \textbf{fire resistance} & & \\ \hline \text{level 14} & \textbf{turn fire/lightning damage into healing} & \text{beasts & plants hesitate to attack} & \text{various immunities} \\ \hline \end{array} \$
It looks like the Circle of the Flame has the best features at each level: best cantrip to allow ranged combat, best melee-buff, extra attack is sick, enhancing concentration-based spells is unique... Even the last feature, that appears situational at first, becomes cheesy once you start to fuel it with your own AoE spells.
Suggestions
So, how do we fix this?
First, you can't aim for balance, and be a strong healer, blaster and tank at the same time. I removed the healing capabilities, as the tank part seemed your main concern, and the theme calls for improved fire damage.
Second, on the AC question: "no metal armor" and bounded accuracy make things difficult. If your DM rewards you with powerful nonmetal armors, he shouldn't allow big AC temporary bonuses in an homebrew class. The opposite is also true: if he allows big AC temporary bonuses... he should make sure you keep a lowly hide armor for your entire career. Hence the solution offered here - which seems the good way to reliably get good AC, without over-investing in dexterity or "forcing the hand" of your DM.
Here's the result:
Leaves choices, and includes another fire-themed cantrip. Green-flame blade is mostly aimed at tier 2+ melee: extra attack was too much (especially if you intend to add Polearm Master), but this cantrip offers something comparable to the cleric's Divine Strike.
Temporary HP is probably the way to go to add survivability to a druid: it matches what has been done for the Circle of Spores and Circle of the Moon. I removed the AC boost: if you insist on keeping it, you'll have to either keep it low (like +2), or remove the ability to use a shield - if you care about bounded accuracy, of course.
With your average 14-dex druid, it means up to +3 AC - and an easier access to magic armor in the long run. It is a game changer for most tables, where nonmetal armor is scarce - and can be balanced by making the ritual as simple or as complex as you wish (involving time, gold, crafting skills, a risk to destroy the item?).
Idea is to open that damage buff to more options (e.g fire bolt, green-flame blade) - while keeping it at a reasonable level (only one creature per turn).
Either affect both targets of green-flame blade, or use it on both flaming sphere and your cantrip of choice.
I'm absolutely not sure that this is well balanced:
But I do think it is closer to "balanced", when compared with Land & Spores druids. Obviously, feel free to switch things around (access to metal armor as soon as level 2?) - and take or leave whatever fits best your idea.