Trying to login to my test site to get the ball rolling on our move, and I've had a new phone since then which doesn't have the site on it. How do I login?
It says I can use a backup code, but I don't have this either.
For 2FA I'd suggest that you use Authy. It works across multiple devices and unlike Google Authenticator it allows you to make a backup so that when you change phones you can restore all of your accounts used with Authy.
Which is why it's important to make sure your server time is correct.
Oh and your point isn't relevant to this discussion which is about resetting logins when you don't have the original phone you used for authentication.
DST doesn't affect the underlying time in GMT/UTC. It just adjust the relative time around that. So no DST issues (as long as it's done properly -- it should generally happen automatically based on the selected server time zone).