The British relationship with the rest of the EU was, at least by the time I moved to Britain (so, 2006), adversarial (win-loose) rather than cooperative (win-win), the latter being more common in the rest of Europe, especially smaller nations.

Also keep in mind that this was following Cammeron’s demands for Britain to be allowed to limit Free Movement from the rest of the EU into Britain but not the other way around (i.e. Britons still kept that right, just stopped reciprocating it) “or else we’ll have a Referendum on leaving the EU”, which was what triggered the Leave Referendum, so “We should stay in the EU and change it from the inside” back then as justification to Remain rather than Leave sure sounded a lot like suggesting that the objective of priviledged treatment for Britons over the rest should be better pursued inside the EU.

It certainly didn’t came across as an appeal to it being better to carry on cooperating for the common good that to leave, which would be the kind of thing more likely to sway, say the Dutch or the Portuguese.

I suspect you might be confusing a local cultural artifact you see all around you in British Modern Culture with a general way of behaving in Europe: everybody is selfish to some point, but from the countries in Europe I lived in, Britain was the one were selfishness and adversarial approaches (you have to win, otherwise that mean you lose) were the most common and accepted. IMHO, Britain is in this in the middle between Continental Europe and the US.