There's been kind of a de fact trans-Atlantic relationship the whole time, considering it was mainly Europe that colonized the New World. Then in addition to the cultural similarity, you have all these infrastructural things like the Monroe Doctrine, Five Eyes, NATO, etc. Even the financial system follows a similar evolution that goes all the way back to the fall of the Western half of the Roman Empire, with its furthest extent now in NYC, and London and Germany aren't far behind.
It might be in Europe's interest to take advantage of closer ties with Russia and also being a main Belt and Road endpoint, but I'm not sure where the balance lies there exactly. The EU has always been something of a tenuous prospect with a good deal more internal tumult than you'd get among states in the US, and it's easy to imagine that internal divisions would only be exacerbated by trying too hard to tie in to Russia and China. Plus the US would not want to see that sort of a potential power shift probably unless it were looking specifically to cause foreign chaos to maintain American Primacy.
Interesting idea, but it flies in the face of decades (or arguably centuries) of geopolitical thinking, and it's easy to imagine Europe just getting more cucked as Russia and China both seek to expand their power and influence rather than try to all hold hands and sing songs.