BTW, when I try to login without providing a phone number, I get a "critical security warning" email sent from a no-reply@accounts.google.com address to which I can't reply. Does Google have an email address to which I can explain the situation?

You can try doing it with vumber - cheaper than a sim

Sometimes it does not ask for a phone number at all, depends on the account risk level. I set up Search Console on a fresh Gmail once and it let me in with just email verification. If it keeps forcing the phone step, try creating the account from a normal browser session with a clean profile and add the site directly through DNS verification in Search Console. That part usually works without tying a phone number to the account.