Ok - I'll imagine there are no systems at all right now; starting from scratch. I think to start it'll also make it easier for me to think of it as if it were literally a video game, so I'll do that.
It'd be an Open World game, of course, with highly customization characters. But focusing on the narrative in the style of the gameplay loop:
For StudentsStudy and Grow -> Face Challenges -> Interpret Results and Deal with Consequences -> Study and Grow
For Heroes
Protect People -> Enemies Attack -> Deal with Consequences -> Protect People
For Villains
Interfere with Other Characters -> Attempts to Pull Something -> People Attempt to Stop Them -> Consequences -> Interfere with Other Characters
For Vigilantes
Develop Philosophy -> Act on Their Philosophy -> People Attempt to Stop Them - > Consequences -> Develop Philosophy
(I think that aside for the top-level players and big events, consequences would be personal and not world-changing.)
Looking at these loops, I'd guess these are the things worth rewarding? Training, Fighting or rather Acting, Interfering, and some form of Introspection?
On the note of you saying not to make the system too unwieldy, I'd say the focus would mainly be on the Students or on the consequences players suffer from their actions.
Starting from the ground up, I'd want players to have two categories of stats. Their own body's stats, things like Physical strength, speed, endurance and whatnot, and then a set of stats for their Quirk's. I do think it's important to not just have a stat(s) for Quirks, because a lot of characters will be training everything else too - only a small amount, or none even, of the character's would only be improving their Quirks. Having more stats for characters to specialize in would give more characters a chance against stronger Quirks.
For the game, I'd say the focus would almost entirely be PvP. While it could present some logistical problems at first, the end goal is to have every single enemy be another player. As for making more strategic or twitchy, I'd say both: in the same way that in some stealth games you can barge in or sneak in - players should be able to choose their path through combat.
Ultimately, what I want players to do is grow their characters. What I think that entails is either following the established "loops", or breaking them.
Edit: I guess one more thing I can talk about is my own love for games. As it might be obvious, I love stats and progression in games. Yet my favorite game is the Witcher 3: a game with only light, mostly out-of-sight stats. Not sure what that means, but I figured it was worth mentioning. I suppose I can open my mind a bit more to some other types of systems.