While one certainly could justify looking for stunt+ custom powers... I'm a fairly firm believer in "If you ought to have stunt X, take stunt X." Then again, this may be a result of having to deal with a player who looked at everything in terms of power-gaming efficiency and refused to take any stunt/power/whatever that wasn't "optimal", even if it was something that the character really ought to have. (Like claws. He refused to take claws, preferring to use sponsored biomantic magic to try to grant himself claws for a scene at a time.)
The other thing to keep in mind is: if it's a Power, there are more ways to shut it down. Thresholds, magic circles, etc, all give the GM reason to hold up a fate point and say "No, sorry, your powers aren't available here." Stunts don't have that drawback.