"Make a Ranged Weapon Attack" does not imply a particular attribute. A javelin thrown with strength is still a "Ranged Weapon Attack". The Thrown property tells you what stat to use.
Thrown. If a weapon has the thrown property, you can throw the weapon to make a ranged attack. If the weapon is a melee weapon, you use the same ability modifier for that attack roll and damage roll that you would use for a melee attack with the weapon. For example, if you throw a handaxe, you use your Strength, but if you throw a dagger, you can use either your Strength or your Dexterity, since the dagger has the finesse property.
As odd as it may seem, a Javelin is actually a simple melee weapon according to p149 of the PHB. That means you use Strength. It does not have the Finesse property, so you may not use Dexterity.
The type of save required by the lightning bolt effect has no bearing on how the javelin itself is thrown. If it did, it would say so. It is worth noting that the target doesn't make a save - the wielder and the target are specifically excluded. The target takes the lightning damage on a hit.
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.
Best Answer
Unless the spell states that you need to make an attack roll, you simply cast the spell and the target(s) must save against the DC of the breath weapon or spell.
So if you have a creature with a breath weapon that has a con of 16 for +3 and +2 prof bonus, you use the breath weapon and all targets must roll a save vs 8 + 3 + 2 = 13. They then take either full or half damage.
Eldritch Blast is an example of a spell with an attack roll and there you roll an attack vs targets AC if it hits damage is dealt.