I'm trying to answer everyone and you pointed out correctly something I didn't define well in my original post: I was trying to find either "believable" powers (in the sense of being well constructed) or "believable" origin stories. They didn't need to come from radioactivity only. The reason I was excluding mutants in my original post was that they have powers since they have a different gene, but that is a very "cheap" way of creating a superhero, since no other explanation is necessary!
Wolverine (as pointed out in the first answer to your comment) is born a mutant and later on given an indestructible skeleton.
Make no mistake: the post is not about superheroes being or not being cool because of their origin story or super powers. I really like Wolverine and Sabertooth!
That's an interesting take, because it is a plausible origin story arising from an "implausible" one! What I mean is that, according to the (admittedly not well defined) "bounds" of my original post, mutants have a less believable origin story since they're just born with powers, but Deadpool becomes a superhero since he already lives in a universe where mutants and healing factors exist and so it makes sense they make experiments on it!
I didn't know Norman Osborn tried to kill Deadpool with a cure for cancer. That's a neat plot device!