For funsies, I went back and reviewed the one episode of Anything You Can Do, at least the only one I have. And yeah, it's Beat the Clock, except that instead of an arbitrary clock, a team of men and a team of women each compete in the same stunts and their times are compared to each other. At the end, the team with the lowest total accumulated time is the winner. Also, there's never a visible clock of any sort, so we're left to take the word of Gene and his offstage folks.
The games I saw were maybe mildly embarrassing, but hardly dangerous. Gene left because he had the chance to actually host Beat the Clock, a much higher-profile gig than this one, and every bit as "dangerous".
BTW this episode is Exhibit A as to why Gene is famous as an announcer and not as a host. He's really just this side of inept. He actually crosses over the ineptitude line at the end, when seconds after announcing the winning team, he congratulates the wrong ones.
