In more recent times they've had Hyperion, who's pretty much a point-by-point ripoff of Superman down to his origin and alliterative secret identity; and the Sentry, who was created to be an over-the-top throwback to the Silver Age.
Didn't they also have Gladiator, the alien Shi'ar? He had Superman level power and was all stuffy and uptight.
And Michael "Mister Terrific (2)" Holt, while motivated by a pep talk from a supernatural entity after a great personal tragedy, is also self-made and pretty sane.
As was the original Golden Age Mister Terrific, Terry Sloane, though I don't think Sloane got any supernatural pep talk. He was just really really good at, uh...everything...and decided to Do Some Good.
Same with the Golden Age Atom, who was a five-foot tall wrestler type. Later on he got super-powers, but originally he was just in it to Do Good and, incidentally, beat up people. There may have been some Short Guy Issues, but I don't think that would qualify him as nutso.
I think an interesting case is Rex Tyler, the original Hourman. He was just a chemist who stumbled on a super-vitamin formula, Miraclo, and used it to fight crime (it gave him super-strength and speed and nigh-invulnerability for, you guessed it, one hour). Later on he became addicted psychologically to the stuff, but that's a case of the crime-fighting making a psycho, not a psycho deciding to fight crime.
Alot of the original mystery men were just dudes who decided to dress funny and beat up criminals. Mister America (later Americommando), Manhunter (I)...the original Red Tornado was Ma Hunkle dressing up in long johns with a pot on her head.