Your analysis is correct but one sided

You have accurately pointed out many of the advantages of the Moon Druid. It is true that it is one of, if not the strongest subclass in the game. But you have failed to account for some of the drawbacks that limit their output.

• Low AC: Most beast forms have comparatively low AC compared to the martial characters in the party. The brown bear you have based most of your analysis on has an AC of only 11. Most attacks are going to hit so you need those HP to make up for it. Additionally you will be making lots of concentration checks for any concentration spell you cast before Wild Shape. Barkskin can help somewhat but AC16 isn't that high and it means you can't concentrate on spells like Moonbeam or Flaming Sphere instead.
• No casting in Wild Shape: Yes the Druid is a full caster, but they have no access to this while in Wild Shape. You have a lot of options and heaps of resources but are limited by the action economy to only doing one thing at a time.
• Weaker Spell List: Compared to the Wizard/Sorcerer spell list the Druid has a lower damage output for pure spell casting at most levels. Then compared to the Cleric they have access to fewer buffs and high level healing spells. I'm not suggesting this makes them weak, but they certainly aren't the most powerful full caster going around.
• No decent cantrips: When the Druid is casting, they have to use spell slots to get a decent damage output. Their cantrips are very weak compared to other full casters and they have no options to add their casting mod to it unlike clerics, warlocks, or wizards.
• Can't use combat feats: The Moon Druid is a strong melee character while in Wild Shape, but not as strong as Fighter or Barbarian. They have a low AC and can't access feats such as Great Weapon Master or fighting styles to increase their output.
• Complexity: Playing a Druid can be overwhelming. With the wealth of options and abilities available it can be hard to make effective choices. It is not a class I recommend to new players. Unless they are really keen on the Wild Shape part.

Experience

I am currently playing a Drow Druid. In my party (level 6) we have a Champion Fighter, Rogue Scout, War Cleric, Evocation Wizard. I have been tracking the kills and damage output in combat since the start of our campaign. Early on, I was slightly in front as Wild Shape gives a huge bump at 2nd level. As the campaign continued however the Rogue quickly overtook me as she increased her damage output with additional sneak attack dice. After gaining Extra Attack at 5th level the fighter has also surpassed me in damage output. At 6th level and gaining a magic weapon the fighter in by far our highest damage dealer.

My typical combat strategy is to combo Moonbeam and Brown Bear Wild Shape (Polar Bear at 6th level). This is pretty good until I lose concentration, which usual happens in 1-2 rounds due to my low AC (lots of checks) and no proficiency in Con Saves.

One session our Cleric was missing and I was forced to drop Wild Shape to heal my allies. This is evidence of the "can't do it all" limitation of the 5e action economy. While I end most encounters with more HP or spell slots than most of my party I'm not able to use all of my abilities together.

Level 6

This opens up new Wild Shape forms for the Moon Druid for the first time. CR2 forms are a big step up from CR1 and will address the reduced power at level 5. The best damage combat form at this level is the Giant Elk. 4d8+4 damage against prone targets is nothing to be sneezed at. There are other great options such as Giant Constrictor Snake but they all suffer the same low AC.

However you only get one attack per turn and still have a low AC of 14. Meanwhile the wizard now has multiple castings of fireball, the fighter gets a second ability score increase and Cleric get a second use of channel divinity.

After playing at 6th level for a while I can say that my damage output still lags behind the fighter and rogue. This is mostly due to a lower to-hit modifier and a lack of extra attack. The additional hit-points of forms at this level make me a more effective tank, particularly now that I have enough spell slots to burn for healing. Combats usually involve me targeting a single enemy and tanking them out of the fight while the others deal with the rest.

Are they OP?

The Druid is likely the most versatile class in the game. They can be a competent healer, decent melee character or a pure damage spell caster. However they can't do it all at once and will never be quite as strong as a character that focuses on one area.

The Moon Druid Wild Shape is possibly OP at level 2. It is a huge bump in damage and HP compared to what others receive at the same level. It does tail off within a few levels though as other classes gain combat features. The bump at 6th level keeps you on par with other classes but is the last major improvement for a while.

Overall I'd say it is strong, maybe the strongest but not so much that it is OP. Any correction to weaken the class risks making it unplayable by taking away the flexibility that makes it so much fun to play.

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:

1. Ranged options (such as sling attacks, or produce flame) lack range and damage.
2. Melee options (such as shillelagh or thorn whip) lack damage in the long run.
3. AC is often on the low-end - unless your DM is generous on non-metal armor distribution, or you focus on dexterity.

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:

Only You

When you choose the Circle of Flame at level 2, fire bolt and green-flame blade become druid cantrips for you. You learn an additional druid cantrip of your choice.

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.

Heart of the Flame

At 2nd level, you gain the ability to awaken the primal spirits of flame. As a bonus action, you can expend a use of your Wild Shape feature to awaken your inner flame, rather than transforming into a beast form. While this feature is active, you gain the following benefits:

• You gain 3 temporary hit points per level you have in this class
• You shed bright light in a 30-foot radius and dim light for an additional 30 feet.
• You add your Wisdom modifier to concentration checks.

These benefits last for 10 minutes, or until you become unconscious.

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.

Purification through Fire

Starting at level 6, you gain resistance to fire damage. You also learn a purification ritual that allows you to prepare metal shields & armors, so that you can use them.

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?).

Heat of Battle

Starting at level 10, once per turn, whenever one of your spells inflicts fire damage to a creature, you can add your Wisdom modifier to that damage.

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).

Flaming Soul

Starting at level 14, you gain immunity to fire damage. You can also apply your heat of battle on a second damage roll, each 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:

• I feel Nature and/or Forge Domain clerics might feel cheated with this, as some of their core subclass features are included here, with few drawbacks
• having both a better AC and temporary HP, while keeping access to healing spells, means that you are tanking pretty hard
• Extra attack was too much - and this may also be. Green-flame blade + Heat of Battle + shillelagh makes a strong combination.
• Bonus to concentration might go out of control if you go for the War Caster / Resilient (Con) route. Replacing it with Con save proficiency or advantage on concentration checks may help avoid those shenanigans.

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.