Sure it could work. It already exists, as a matter of fact, it's called mixed martial arts.
Professional wrestling without a script would ultimately be nothing more than what MMA ultimately is, which is shoot fighting, and it'd be courting disaster. Look at how many wrestlers could wind up being injured or outright crippled. Also, personally, I find real fighting to be kinda boring. A pro wrestling match is more like some sort of planned, choreographed fight you'd see on television or in the movies. It's designed to look visually great while simultaneously as safe as they can be. Doesn't mean wrestlers don't get hurt or injured, just as can happen in movie or TV fight scenes.
As far as the notion of it being unscripted where the wrestlers do their own promos, storylines and matches themselves, I dunno why it wouldn't. It'd ultimately be removing the middle man since the wrestlers would have to run the ideas by certain members of management just like the writers do. When it comes to a creative process of writing a story, book, poem, planning out ideas, arranging them in certain ways, etc., some people are going to be better at it than others just like some have a better delivery on the mic than some others, how some are more comfortable doing promos than others, how some are better inside the ring wrestling, etc. Is that really any different than a team of bookers/writers doing it? Not really. In NXT, Triple H has the wrestlers use their own words & thoughts and the only aspect of a script involved are a few bullet points that the wrestler is supposed to touch on in the promo.