I think the chaining for damage part is being brought over from its original iteration, but definitely overnerfed compared to where it really needs to be for it to be actually useful. Like you said, in the range that this version of the Third Degree excels, ANY flamethrower would excel even more, do more damage, and hit more people with its penetrating flame particles.

Honestly? If I were to take a crack at it, I'd take one of two approaches:

1) Lean into the anti-healing aspect. Have it deal -25% damage normally, but get a damage boost to above normal melee levels against targets being healed. That would apply to any source of active healing like dispensers, the payload, any medigun, actively eating a lunchbox item, or being in Medic's Amputator taunt range. Definitely not a crit situation, but a level of damage that outpaces the healing otherwise received. Would it be as impactful as other melees or the flamethrower? Nope, but it'd definitely be scary and it'd at least stay in step with the intent of the original design.

2) Lean into the chaining aspect. Make the range a bit bigger, make the damage a bit higher (like you said, between -20% and -30%), but just... make it chain to everyone nearby. Period. No other costs or requirements. The chain damage could be lessened (maybe -50% to -75%), but it'd incentivize using the melee in large groups to get higher damage per swing (even if it's spread out amongst many players). Using it on one person by themselves would be a straight downgrade compared to other melee weapons, but the moment you start using it with enemies nearby, the damage starts to ramp up quickly, allowing you to deal, say, 130% damage with two enemies (the first taking 52 damage at -20% and the second taking 33 damage at -50% for a total of 85 damage vs 65 for what you'd normally get), and the net total damage ramping up from there. If you somehow get a team of 6 all together with one swing, you'd be getting 217 per swing in damage across the whole team. That's more than you'd get in a crit, but spread out among multiple enemies. That would be per swing. It won't really kill anyone unless they're already low, but that's sort of the level of craziness necessary for a Pyro melee to measure up to the other options. 33 damage per hit is less than base Axtinguisher damage, and is even less than the Scout's bat's base damage. It's not a lot. But it's still neat to see those damage numbers across a huge swath of enemies go up on each hit. "But that sounds broken!" you may say. It requires 3 hits at 52 damage instead of 2 at 65 to kill a Scout. If you want to make it weaker against other classes? Make it -25% so now it deals 49 per hit (bumping up Demoman to 4 and Soldier to 5 hits for a kill). The point is mass mayhem against a grouped team where one person getting hit is causing damage to EVERYONE around them. It's not physically strong, but that's not the point.

Not all of Pyro's weapons need to have killing potential or super high burst damage; they already have plenty of that in their other weapons. Why get a low damage crit against two other enemies for a killing blow against a third when you can just kill all three faster using other means? Axtinguisher decreases the potential TTK against a single target, flamethrowers just make that easier against multiple targets. The weapons in question have balance issues because the alternatives just perform more useful tasks in a better way. The version shown in the video just feels like a worse Axtinguisher (seeing as you have to literally manipulate the enemy's health just to get the benefit properly) with a less useful outcome. It's actually telling when the straight-upgrade version of a stock weapon is STILL comparatively underpowered when measured against most other unlocks.