Gmail SMTP server going offline

Are you sure google isn't simply kicking you out after a month or so, because you have to refresh your two factor auth login?

SMTP uses password authentication, and two factor auth sets up your account so that a password alone is not enough to access your account.

I think this is unlikely to be a problem with mail.app, and deleting/re-creating the account is a red herring. The real issue is happening on google's server and something else is solving it. Deleting the account in Mail.app and creating a new one with the same username/password has no impact at all on what data it sends to google's server, so it's impossible for that to solve the problem you're having.

Perhaps if you log into google in safari, go through the two factor auth process, Mail.app will suddenly start working again? I'm not sure how google's two factor auth uses, I think if you have used two factor auth from that IP address recently, it will accept password-only authentication.

Failing that, you might have better luck on a google specific support forum, or just waiting for it to be fixed (you can't be the only person experiencing this issue).

I'm doubtful this is your issue, but note that the password for the SMTP server is configured independently from the incoming (IMAP) server. Double-check both are set to a valid app-specific password in Mail.app's settings.

FWIW I've been having this problem with my account since setting up two-factor-auth. I've had to reset my app-specific password for the GMail account on my mac profile several times. The most recent time I actually just needed to restart Mail.app though.