I am probably missing a whole bunch of names, but maybe it makes sense to look at the "dozens" of German players in the PL over the last two decades or so, to see who'd actually would've suited Man Utd.
- Ballack - would've been great, but moved to Chelsea when the big wall of cash came in;
- Emre Can - wasn't a big player when he moved from Bayer Leverkusen. For his best position, Man Utd brought in Di Maria, considered a much better pick at that moment;
- Gnabry - came to Arsenal as a youth player, and only 1 out of XXX youth players really turn out so great (remember Janusaj?);
- Gündogan: yeah, could've been a great signing. But: in 2016 moving to a strong City or a failing United?
- Havertz: wouldn't fit the current line up, would prefer Bruno;
- Holtby: not even good enough for Tottenham;
- Huth: not a world beater, failed at Chelsea;
- Karius: yeah, no;
- Lehmann: probably would've been better than Howard in 2003. Then again: V.d. Sar wouldn't have joined in 2005, had Lehmann joined. I consider that a win, looking back;
- Mertesacker: when he joined Arsenal, Rio / Vidic were the designated defenders. No contest;
- Mustafi: not Man Utd material;
- Özil: I think this could have worked in 2013, but then again: can you have a team with both Rooney and Özil starting?;
- Sané: I would've loved that. But like with Gundogan: why join United over City in 2016?
- Schürrle: potentially could;ve been a great signing in 2013. He didn't become a world beater, but was seen as a great talent back then;
- Werner: Getting him would mean stopping the development of Rashford. Potentially could've been great, and possibly a better way to spend the Donny v.d. Beek money;
So all in all: I can only think of a handful of players where Man Utd really missed out on, and there are logical reasons why that never came through. The amount of absolute world-class Germans in the PL has always been fairly limited. Just like there are not so many brilliant English players in Germany / Spain / Italy...