I can use Safari, then Google login, then get to any of my Gmail accounts, so I was finally able to get to the email telling me of your reply.
So using "webmail" works.
Just a reminder that this problem has started very recently and I have no new apps. I don't recall exactly when I updated to 12.4.
So, when I go into Mail it shows no gmail accounts connected and when I try to connect to that same account that I got to through webmail, or I click on that account in Settings/Internet Accounts, it tells me "Google requires completing authentication in your web browser", which then fails when I click on Open Browser. It won't even let me add a new Google account to Internet Accounts, telling me the same "Google requires ..." and then resulting in 'Authentication Failed. Failed to authenticate with Google, please try again later."
Of course, by now, I've "tried again later" many, many times.
While the problem impacts Mail, it appears to be rooted in a problem with Internet Accounts if it won't even let me try to start adding a Google account - I never even get to an email and password request screen.