Just an update for any other US citizens who might need to renew their passport... I just finished renewing mine (that was set to expire in April). I followed (exactly) what they said
here for a renewal by mail. Going to the post office to fill out the money order (still called "Postanweisung für das Inland") was the most trying part because I don't speak much German and handing over that kind of money "hoping" I had the right form was a little stressful.
But I just received my self-addressed stamped envelope back and inside was my newly minted passport (along with my old one). I sent off the paperwork on the 26 or 27 of Jan and received it back in the mail 10 or 11 of Feb (had to pick it up at the post office today with an ID). So it didn't take the minimum 20 working days they claim on the website, but I am assuming it is because I did it right after Christmas and well before summer, so I am in the height of slow season.
Good luck if you do it too, but it was pretty painless for me.