And, technically, oral contracts can be valid and legally enforceable in many contexts -- assuming the trier of fact later finds that the oral contract existed. "He said, she said" is not a rule of law. It's just a reflection of the practical reality that if it's somebody's word against someone else's, it'll come down to who the trier of fact believes, with the plaintiff in a lawsuit typically having the burden of proof and persuasion on most issues. (And it's also a facile way for the police to sluff off a citizen's complaint/crime report that they don't want to pursue/prosecute. Trust me, if they want to pursue a prosecution, the fact that it's a "he said, she said" situation won't stop them from doing so.)
But, for this Survivor situation, it definitely would be foolish to not have a written contract spelling everything out, including such persnickity details like choice of forum, venue, and choice of law for any later dispute -- where you may well have chop participants spread across multiple states.
And they have limited time each week to herd all the cats to make a final agreement. We're already now in the second game of this NFL week, and their selections are due I believe at 4PM PT today.
This contest draws a lot of attention because of all the win-or-go-home drama involved each week, but to me it's like buying $1000-a-pop lotto tickets. I'd rather put the money into the side handicapping contests where I have a somewhat better chance to win -- such as that is. And I certainly wouldn't want to be dealing with all the distraction while trying to pick five side winners against the contest numbers.
Great drama to watch, but I wouldn't want the distraction from what's already a tough enough nut to crack in the side handicapping contests.