Is this balanced against original Barbarian paths? No. Not at all.

Let's take this level by level:

Level 3: Building Charge

First off, it's unclear as to how the static points are generated. Is it just if there's an enemy in range? Is it if you strike an enemy in range? Regardless of how the points are gained, at level 3, they can have a maximum of 3 points, meaning they can expend either 1 or 2 points to use an action to shock an enemy. This means they'll likely be taking 1d8 or 2d8 lightning damage, which, at earlier levels, seems on par with default barbarian damage.

However, compared with the level 3 abilities from the Player's Handbook, on pages 49 and 50, we run into an issue that will continue to come up throughout this path: the other abilities tend more towards support abilities than damage. The Berserker gains the ability to go into a frenzy and take an attack on a bonus action, but with the harsh penalty of gaining a point of exhaustion at the end of the rage. The Totem barbarian gains either resistance to more damage, more mobility in combat, or an aura that helps nearby allies.

I think at early levels, this ability might not get used; I'm not great at statistics, so if someone has run the math and can tell me I'm wrong, that's fine, but I believe that for a while, this ability will be worse than the barbarian just attacking normally. Once you start reaching higher levels, this starts to be a lot of d8s of damage that the barbarian can put out. And, notably, this is a magical form of damage, which means that the barbarian will be able to effectively hit monsters that are immune to piercing, slashing, and bludgeoning damage. Whether or not this is unbalanced depends largely on the campaign, but it's worth noting.

Level 6: Fly like the Wind

Meh. Flying is incredibly situational. However, the Totem path doesn't get flying until level 14, so this is still weird. Getting flight at level 6 when most classes can't get it until much later is odd.

Level 10: Lightning Strike

This is similar to the Paladin's "Improved Divine Smite" ability, which is gained at level 11, but also does 1d8 damage as opposed to the 1d6 for Lightning Strike. So, compared to a Paladin, this is okay, but again, it's very different from the PHB options. The Totem path gains the commune with nature spell, and the Berserker gains the ability to inflict the frightened condition, both of which I would classify as "support" abilities, rather than damage dealing ones. This is a decent chunk of extra damage, and at level 15, it'll actually outpace the Paladin's Divine Smite.

Level 14: Electric Discharge

Absolutely not. At level 14, this is 7d8 lightning damage to everyone nearby. At level 20, it's 10d8 lightning damage to everyone nearby, in addition to the fact that at level 20, you have unlimited rages. AOE damage is an enormous gamechanger; there's a reason that AOE damage is pretty rare. Having it be once per long rest helps, but it still doesn't bring this on par with the other Barbarian options at level 14. Again, they don't add damage to the class, but rather tactical and supporting abilities. This is the appropriate level for flight, for example, in the PHB.

Finally, regarding intent:

You say your goal is to have the barbarian be able to do damage at range, but I don't think this will work well for that. The Stormy Aura requires lots of targets within 10 feet in order to charge up points, which I wouldn't consider as being at range. Additionally, the point of the Barbarian is to be in melee. That's why they have such high constitution: they're the front line tanky people who stand there and get hit so that the squishy rogues and wizards can toss daggers and spells around without fear.

I think this is a really cool concept, but I think it's weirdly balanced throughout levels. Compared to the original barbarian paths, the extra damage output of this is just too much, especially when the barbarian paths contain primarily support abilities, and the few abilities (like frenzied rage) that grant extra damage come at a high price.

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.