It's tough just because most of the well-known announcers are ones that are generally the old school same type of person.
I don't there's been any bad choices made, but in general you're going to end up with most of the classic players just because of the group here.
That being said, Burton should at least be in consideration here given how prolific he was in the 90s into the 2010s. His voice was distinct and fit in with a lot of shows, whether it was To Tell the Truth, Family Feud, Family Game Night.
I'm struggling to come up with any good women who'd be viable for an announcer Mount Rushmore. There's been good ones like Debra Wilson on Weakest Link and Loretta Fox on Rock and Roll Jeopardy, but none of them has had a prolonged and storied announcing career as most of the ones being named.
I feel like it'd be most of the old guard for me on this list too. Gilbert, O'Donnell, they were part of multiple game shows that have stood the test of time and Gilbert's really the last of his kind, not only in a living sense, but also still announcing and doing things for Jeopardy. I'd probably put Burton there too to make sure the newer generation of game shows get represented. As for a fourth? I think it'd be one of Rod Roddy or Johnny Olson, where Price is the big lengthy, but both had big stuff outside of it too, Match Game and Body Language for Olson and Press Your Luck and Love Connection for Rod Roddy.
I think they're both distinctive enough and have their own flair that I'd put them both up there (though if you must limit it to one, Johnny by a hair).