I have every class at max level, but am far from an expert in them all. During Legion I got all the appearances for every spec, but didn’t get Spelltome in Shadowlands (got it last patch here in Dragonflight). Some of my assumptions could be wrong since balance is different both times, but:
Thwarting the Twins: I consistently got to the end with Frost Mage, but had trouble sealing the deal. Eventually I switched to Shadow Priest. I thought I would have more trouble since they have fewer slows, but Shadow Priest let me power through it super fast. Frost Mage has since gotten a substantial rework though, so may come out ahead again.
End the Risen Threat: I used Restoration Shaman, since they have very good (and very easy) damage for a healing class, and lots of good control options for the first phase of the scenario. This was prior to the Holy Paladin rework, and I feel like Holy Paladin is now the right choice.
Highlord’s Return: I used Guardian Druid. I think Vengeance Demon Hunter is probably fine as well. Basically the only thing that kills me here is letting Velen die (entirely my fault) or getting knocked off the edge by infernals (I’ve died more to this than anything in any scenario). Both of these two classes are very good at nullifying the infernals’ knock-off ability, with wild charge or glide.
Feltotem’s Fall: Windwalker Monk. Don’t even try anything else. Monk makes it a joke compared to micromanaging CC with a pet class.
An Impossible Foe: I used Elemental Shaman, because it seemed like you could spec into a lot of medium-cooldown damage abilities that lined up very well with imp spawns, and because I was relatively comfortable with shaman already. This is a very tentative recommendation, since I’m not an expert on any of the other relevant specs that do this scenario, and didn’t spend much time on it after trying Elemental. Of note, Fire Mage is apparently broken after their rework, and may be the best here, but I did it before the rework and barely play mage.
Closing the Eye: I used Arms Warrior. Spec the character to have a lot of powerful, medium-CD AOE damage abilities to deal with crystal prisons in first phase and voidwalker spawns in second phase. I had issues at the end killing the boss before the room filled up with ground effects, but my rotation was bad because I never play arms, and I kept messing up the positioning of the pools. This is also a tentative recommendation.
The God-Queen’s Fury: My main is a paladin (usually holy) so I picked Retribution for this one. I am extremely biased because of this, but it felt like as Ret I was able to mostly just power my way through without having to worry about mechanics as carefully. I kept hitting a point in the fight where the boss got off her super-ultra-major attack, but I could swear it happened without being interruptable for some reason. I don’t know if I came up against a timer, or if it had something to do with the henchmen dying, or if I just consistently missed a cast late in the fight. But I found that, as long as I was built for survival and had all my cooldowns up, I was even able to survive one (1) cast of this ability. First using damage reduction and hard-healing myself, then bubbling when I got low and continuing to heal, then using my explosive shield when bubble wore off and continuing to heal until her cast was done. After that I had to go all out to kill her before the next cast, and managed after a few tries.