More specifically, I would LOVE if the long term dynamic became more of equal partnership and division of concerns - up to and including mutual basing agreements on BOTH sides of the pond.
We have Ramstein and Incirlik and Lakenheath and a bunch of others; maybe it makes sense for the EU to have a few disused bases in the US too - some joint, some just for them. Something in Texas, Alaska, somewhere in the PNW, East coast somewhere, Florida, and Guam would make a lot of sense, I think, even if (outside of Guam and Alaska, for somewhat obvious reasons), I would think they’d be largely training focused - but I think that sort of thing would be extremely helpful in terms of strengthening the alliance, and making the EU as a whole a much more obviously equal partner.
The only valuable US territory for EU nations to host troops would be Alaska and the Pacific territories, and Alaska would be the only decent location for European defense. Any other locations would only be useful for power projection, in which the EU seems rather disinterested in.
That said, the US hosts a lot of training centers for EU troops, including hosting a ton of pilot training schools. So there are a lot of EU troops in the US, they are just on American bases.
Unofficially : US doesn't really want that because that would significantly reduce their diplomatic influence and weapons sales. European nations don't want it, because of responsibility, it would be expensive and we would end up with a lot of armed nations that don't really like each other that much.
So, it's really beneficial for everyone that it stays like it is.
Unofficially : US doesn't really want that because that would significantly reduce their diplomatic influence
The US has been pushing a lot of it behind the scenes for several Presidents. As the world's economy becomes more equal, the US can't afford being the only country providing the bulk of defense. There have been some minor disagreements on coordination, but I feel like these disagreements have been inflated.