I think Europe and the USA sadly are drifting apart.
In a few years the people with Euopean ancestry in the USA will be a minority.
The new U.S. will look to latin America and Asia...
Nothing is set in concrete. An independent UK would have a lot more flexibility to develop a closer arrangement. Why not negotiate trade and visa arrangements with the US rather than Europe?
As I noted above, people tend to limit their thinking to the current reality rather than what could be.