Even "sehr geehrte Müller" already includes the gender (it would be "sehr geehrter" for a man), so if you want to avoid the whole situation, you have to go informal, and indeed, I have received a lot of emails recently which start with simply "Hallo [firstName] [lastName]" (which I guess is still more formal then only "Hallo [firstName]").

Sure, but in a more formal setting you wouldn't address somebody with "Hallo Martina". So my question is how would you avoid the Herr or Frau in a formal setting?

It turns out the people who want to be addressed by neutral gender usually don't care as much about Siezen and honorifics, so the systems I built usually just go with "Hallo" or "Moin". I even had one system which would go with "Sehr geehrte/r" for people who had chosen a title or set their language preference to "Deutsch (Sie)" and with "Moin" for everyone else.

Moin is not a good way to address your Ausländerbehörde case worker though. I'd go for the more disarming Hallöchen Popöchen.

Is it not a similar issue in the US or UK? In a formal setting, e. g. communication from a company representative to an unknown (except for the name) representative from a different company, I would be using "Dear Ms./Mr. <last name>". If that is the correct way (please let me know!) you have to commit to one of only two genders and sometimes also the last name is a guess.

You could say Dear Firstname Lastname without being off.