Being part of the EU by choice isn't really losing one's independence anyway -- not least because we can, and have, chosen to leave, which was an independent decision. So it seems to me that there's already a contradiction inherent in your views on the EU. More than that, though, I suspect that your definition of independence is quite simply nonexistent in the modern world anyway -- which would make you doubly wrong. So, put bluntly, I think you're the one failing to understand "independence".
In the hallowed past when all this was just theory, some passionate Brexiter sent me emails in response to a letter in the Times I had published, assuring me that, being born well into our membership of the EC/EU, I had "never known what it was like to be truly free", which is just the sort of hyperbole that turns me right off the extreme examples of Brexit support. I would love to know how anyone can truly believe that sort of crap.