First, he specifically said "every institution by default expects you to have gmail." Not requires a known domain. So I asked for an example.

Regarding your experience, I'd say that's still very, very rare. Certainly not "every institution" like OP claimed, and probably not even common large institutions. I'd be interested in an example of even one of those. Rejecting specific temp email domains is certainly something some sites do, but as someone that has a couple unique domains that I've used for years, large companies requiring Gmail or rejecting custom domains is not something I've experienced thus am curious about a specific example of such claims.