You do not have to pay a stunt to have superpowers, although your stunts can improve and specialize the powers that you do pick.
For example, a super-strong character might have a stunt like this:
Because there's nothing I can't lift, I get a +2 to Forcefully Overcome an extremely heavy physical obstacle by lifting and moving it.
Stunts can be used to differentiate characters with the same sort of power. One super-strong hero prefers to lift and carry things, and buys a stunt that reinforces that. Another might have a bonus to Create Advantage in combat by using "brick tricks" such as punching the ground to create tremors. If you have a "signature" ability, a stunt is a good way to bring it to the foreground.