I think it's entirely dependent on how "low-key" a hero is. Like, Hal Jordan is a space cop, Thor is a god, Wonder Woman is... whatever her origin currently is. These are people who run in circles that don't need protection, and that are beyond mortal law.
But Spider-Man? Dude's a normal broke dude that occasionally fights thugs. He can't protect the people close to him if a gangster or super-criminal wants to put the hurt in. Even for people like Batman, who have the means, there's the legal issue. You can't run around assaulting people. There's due process.
Like, Tony Stark? Should be in jail. Imagine if Elon Musk strapped on some body armor and started running around shooting people, judge/jury/executioner? We're living in a society here! There's rules! And don't get me started on SHIELD, shit's just unconstitutional...