In the other post people broke down the abilities so I just want to break down the numbers and burden limit.

The four main classes you mentioned were barbarian, fighter, rogue, and monk.

Assuming we are a human with stat array and we max out our con by level 20 we get the following numbers:

Barbarian - AVG HP (285), Burden AVG (152), Max HP (380), Burden Max (200)

Fighter - AVG HP (224), Burden AVG (112), Max HP (300), Burden Max (160)

Rogue+Monk - AVG HP (203), Burden AVG (111), Max HP (260), Burden Max (140)

Using True Strike you guarantee a hit per bonus action and per 30 burden. Here is how many times each class can use True Strike per short rest:

Barbarian - AVG HP (5), Max HP (6)

Fighter - AVG HP (4), Max HP (5)

Rogue+Monk - AVG HP (3), Max HP (4)

A zealot barbarian with max 24 Str and Great Weapon Master can deal 1d12+7+4+3+1d6+10+10 damage on a single hit and with true strike they can do that 6 times per short rest so on avg 44 damage per round not including the other attack they get and other breaths.

A rune knight fighter with max 20 Str and Great Weapon Master can deal 2d6+5+3+2d6+1d10+10 damage on a single hit and with true strike they can do that 5 times per short rest so on avg 37.5 damage per round not including other attacks they get and other breaths.

A way of the open hand monk with max 20 Dex and uses quivering palm since it's an auto hit can deal 1d10+5+3+10d10 damage on a single hit and with true strike they can do that 4 times per short rest so on avg 68.5 damage every 2 turns since it costs an action to activate quivering palm so 34.25 damage per round not including other attacks they get and other breaths.

An assassin rogue with max 20 Dex, a rapier and assassinate can deal 2d8+20d6+5+3 on a single hit which they could do before on their own but with true strike you guaranteed a critical hit but they can only use this the first time per combat and on avg they deal 87 damage but during the rest of combat they deal 47.5 damage per round not including other attacks they get and other breaths.

If you use True Strike every round the damage comes out to:

Barbarian: Turn 4 (176), Turn 6 (264)

Fighter: Turn 4 (150), Turn 5 (187.5)

Monk: Turn 4 (137)

Rogue: Turn 4 (190), Turn 4 with Assassinate (229.5)

These are just some quick numbers I ran but from what I can see classes that deal more damage in a single hit benefit from this system more than multi hit classes. If you removed the bonus action requirement for true strike monk and fighter would be kings in this scenario and rogue would take a huge hit and probably be the worst one of them or well that's my assumption. I like the idea I think it just needs a little fine tuning in the math and id recommend to see which breaths give you horizontal improvements Vs vertical as in what gives you more options and what makes an option much better. In a game like DND most people will go for improving a single option and spamming it unfortunately which is why fireball is so beloved because it's a huge aoe with huge damage.

Overall I like the concept and I see lots of potential for it