If X-Men abilities were real, the challenge wouldn't have come about in the first place, because there would have been empirical evidence for their existence long ago and Randi would never have needed to throw down the gauntlet.
However, if tomorrow a bald guy in a wheelchair sends in an application, and he really does have Prof X powers, then yes, he would win.
But you're talking about taking a character out of a universe where such powers are not disputed, and putting him in a universe where they are. You might as well ask, in the X-Men universe, would there be such a thing as the JREF Challenge. Psychic abilities are not paranormal in that world, so no, there wouldn't, unless perhaps there were other dubious claims that interested Randi.
If the X-Men story came true, then there would be a 'first' psychic, if he applied he would win because he actually does have the powers. The reason no-one has won so far is that they don't. It doesn't matter if it's because of a mutant gene or sky goblins, the list of paranormal abilities is as stated.
If the gene were identified and proved before any psychics had applied, maybe Randi would withdraw that aspect of it, who knows? As you say, it wouldn't be paranormal anymore so what would be the point?