you are being requested to authorize access using the authentication method oAuth2.0. While there are ideological arguments about using the authentication method, the short answer is it is Googles preferred method, so if you just fill in the googlee password and allow it to continue, your gmail experience will be much less troublesome. It only appears once after each change of password on Google is my understanding.

If you do not use oAuth2.0. You have to make sure that gmail is configured to allow less secure apps and for a non confrontational time you really should not use the two factor authentication, the application passwords required for two factor authentication have a habit of periodically expiring or something. Requiring they be recreated.

Once all that is done you can change the authentication method in Thunderbird to normal password. Both in account settings for the incoming account and outgoing server (SMTP) for outgoing mail.

Thanks Matt. That solves one side of the problem, but I have a second one which is that when I try to click on the next button, it times out as I show in the second image.

I know there's no problem with my internet connection, so it must be either mozilla or gmail. Nor has this solved over time. I gave it a break and came back a few hours later, and the problem is still there.

I have just in the last few minutes solved the problem it seems. I changed the authentication method to "normal password" in the outgoing server settings in account settings.

Things became just as normal. Lesson learned here... Stay away from the oAuth2.0!

You probably have cookies disabled in Thunderbird. My understanding is you have to have it on or the screen never updates and eventually times out as you said.