For me, I think (in context) cool fights are either raw power struggles between two powerful characters, an underdog fight of someone creatively using their less powerful Quirk to win, or something like two people with non-combat quirks having a down-to-earth realistic fight. The bulk of the fights will be context of the school - training, exercises, but predominately tournaments. A lot of these will carry in-character weight - since the placement of characters could affect their desirability post-graduation. (Sounds a lot like the real world there.) Otherwise for non-student characters, these big fights will ever be the culmination of long chases or clashes of ideology. For them, it may sometimes be life or death, the imprisonment of a dangerous individual, or public acknowledgment.
I do know that a lot of different fights will look different: some may be between two characters blasting each other with massive elemental attacks or two sluggers brawling it out. Some will be mind games and some will be calculated strategic chess-like combat.
One thing to note is that this system won't always be applied - that much is left to the digression of the thread partners. They're free to go freestyle as long as they agree - the systems are mostly there for official things like tournaments, and of course, disagreements.
I suppose the most similar system we have/will have to your city control deal is the rankings - the Top 20/10 of each group: Heroes, Students, Villains, and Vigilantes. In the future these'll be determined by a calculation of their stat totals as well as their popularity in the site's Popularity Polls.
I suppose, in my head, the main goals would be for players improve their character statistically and also quality-wise (and socially, more likely). So that'll be what they're doing.
At the moment I think the current system is a good balance in that regard, Serpent: while most of it is simple stat comparisons, players have the choice to dodge, block, use one of their skills, etc. So while it's (relatively) simple, I think it makes for some impactful fights given skills are essentially a player's super moves. (People can create skills that use their Quirk in a creative way to a strong stated effect.)
