There's still extraterrestrial/extradimensionsal/ supernatural threats you can have heroes deal with. Its more science fiction/fantasy than the typical crime stories of the likes of batman, but was and are still common adversaries of superheroes
What could work is to have the Court of Owls be the villains but also have "Batman" be a major collective effort rather than just Bruce Wayne. Instead of Bruce being born rich he could have been a son of a maid who worked for the Waynes and was close to them before they were killed by a hitman hired by the uber-rich for their support of labor rights, he could then be a former soldier of the revolution who learned how to fight and always wanted to be a detective, has Alfred be the former butler for a wealthy Wayne family who has access to their estate and was an actor before becoming a butler and has Lucius Fox be a brilliant inventor who builds all of Bruce's gear. Instead of a one-man show they exist as a big team effort, with new characters being added in all the time.
The big issue with Superheroes in general is that a big concept of socialism is that crime is a result of inequality in class. A post-revolutionary America it would be expected to not have issues like gangs and corrupt cops, though obviously it still would. Yes, you could have the Silver Age idea of just fighting random mad scientists or counter-revolutionaries, but for the most part stories about regular criminal activity due to poverty being widespread would be seen as counter-revolutionary.
Agreed. American superheroes won't really be crime fighters ITTL (European ones might). The exception will be organized crime, which does still exist, albeit on a much smaller scale. The abolition of capitalism both reduces the incentive for people to turn to crime and opportunities to get rich through crime, as the IWW's practice of keeping finances public makes things like embezzlement harder. We also won't get Prohibition, which was the real catalyst for the development of nationwide crime syndicates. But the Revolution didn't abolish organized crime entirely, and straight-up robbery is still a thing.
Teams will definitely be much more common and Bruce Wayne himself will probably be lower class. That said, once comic books start developing more complex storylines, another hero coming from a wealthy background would be likely, with part of their narrative being them coming to terms with the Revolution. I'm leaning towards the daughter of a wealthy family who was a teenager when the Revolution came, and one of her traits is that she can infiltrate high society abroad very effectively. This might be delayed until the Silver Age.
Actually, from my limited knowledge of superheroes, the one who fits this the best could be Moon Knight, aka Miriam Spector, onetime heiress to a wealthy Jewish family. When the Revolution came, she stole a few artifacts her father had in his collection, initially planning on just selling them, but instead they gave her powers (later retcons had them also stopping her aging). Being Jewish, she'd be no friend of the Whites or Falangists, and some writers would suggest that she always had socialist sympathies (this is a major point of contention among fans, with some feeling that it weakens her character development).
