You don't want to play Q&A material against a clock. That's one of the things that killed Whew! You also need to come in with a mountain of material and that requires writers, who don't work for free.
That's how most high-school quizzes do it, especially near the end of the show. As for that "mountain of material", for example, I think "It's Academic" has to come up with about over 7300 questions for all of its versions nationwide in a single season. (Granted, the show might recycle some questions every several years.) Plus, a lot of the production staff in Washington, D.C. also are question writers themselves (they write the questions for all versions nationwide).
Full disclosure: I'm now the honorary historian of the entire "It's Academic" quiz franchise.