> Why isn't there a NATO base in the US where German military etc. "hang out"?

The reason there is a large US presence in Germany is that Germany lost WWII. You can make all kinds of other justifications for it like "Germany invited the US", but it all comes back to the Allies created the current German government, and all of them had troops continuously stationed in Germany from WWII until well after the cold war ended. France and the UK have withdrawn most of theirs, the US is the one that still has a sizable presence and the reason is that Germany is afraid of Russia and wants them to stay.

> ...wants them to stay.

This may be true now, by a small margin, but until two years ago, the big majority of Germans wanted US troops to go away.

I can't even count how many times I saw or heard "Ami go home!"

No? Just because Sweden is a member of NATO doesn't meant they have to just let the US do whatever they want. IF Sweden so chooses to allow US troops to be stationed there, it will be for their own benefit.

Sweden hosts US troops already. There was no reason to wait for Orban and Erdogan for that.

> Why isn't there a NATO base in the US where German military etc. "hang out"?

The same reason that within NATO, German troops are forward deployed to Lithuania and not vice versa; it doesn't contribute to defensive strategy to backward-deploy forces from the countries closer to the large conventional threat to the ones farther from it.

The German Air Force's USA/Canada command has a couple thousand people in the US (granted, the US military has more people than that in Germany)

> Why isn't there a NATO base in the US where German military etc. "hang out"?

Because Russia is in Europe, not in the US.

There is the GAFFTC (German Air Force Flying Training Center) at Holloman AFB?

Most (all?) of German jet fighter training happens partially in the US.