I think those thinking that democracy or any government would collapse are way off base. Huge armies exist today with weapons of mass destruction and are held in check by other armies with weapons of mass destruction.(Super simplified yes I know) The same would hold true in a world populated with super villians and super heroes. They would hold each other in check. Sure you have a Doom in Latveria but you would have a SHIELD or Checkmate to keep tabs on them and keep them in line.
Except that nations and governments have numerous internal checks, balances and innate stability that super-powered individuals(often with extreme traumatic experiences in their pasts) would not have. You have power that in some cases would compare to weapons of mass destruction, and are giving it to people with no accountability or responsibility to anyone but themselves.
I do think a lot of governments and nations would collapse in such an environment. As more and more 2nd, and 3rd world nations realize they can't deal with certain super-powered individuals, the UN would probably be asked to step in. The UN would probably gain much enhanced powers, maybe even becoming a sort of one-world government, complete with its own military and teams of supers to bring worldwide stability.