> and lo, an hour after connection I get a message, that due to strange activity on my account, I need to enter code sent via sms.
It's interesting you got that message (via email?) one hour after you successfully signed in on your iphone. Are you sure it was not some phishing email or something? Also are you still logged in on that account or did you get logged out?
Ah, I see. The only thing that comes to mind is to check if you followed one of the tips from this page: https://support.google.com/accounts/answer/7299973, which is to use a computer (or phone), or same location from where you have successfully accessed your account before ("like at home or at work"). Of course that is not a guarantee of anything, but worth a shot to attempt account recovery again if you have an option to try one of those things.
When you say "I still have access to the email address in question" - are you signed in to the account somewhere? If yes, you could turn off 2FA in myaccount.google.com as long as you remember the password (which you say you do)
It's interesting you got that message (via email?) one hour after you successfully signed in on your iphone. Are you sure it was not some phishing email or something? Also are you still logged in on that account or did you get logged out?