I always felt it was more of a "big country which was once an empire" problem.

Even my tiny Portugal has some level of "celebrating the greatness of the country" on account of what was done by people there 5 centuries ago (i.e. during the time of the Discoveries) and Britain's "Glory Days" are closer in time, were even bigger and, maybe more importantly, unlike in my country were that kind of stuff is out of fashion because the Fascists were heavy users of it, there are frequent celebrations of past glories some way or another in British media and there is often a nationalistic slant in the reporting of international news (mainly of Britain being portrayed as more important and listenned to abroad than it actually is if you look at the coverage of the same news in other countries).

It's understandable when you've spent your life seeing international news being reported always with a slant that makes Britain seem important (which it isn't anymore than other similar-sized countries, and often looks kinda desperatelly trying to ride in the coat-tails of America, the real important ones) and were the Imperial past is constantly remembered (nowadays, mainly with films and TV series), one ends up believing "we're special" and that was easilly weaponized by the Leave campaign and is still weaponized by the Tories.

Most Brexiter arguments wouldn't at all be believable for those with a realistic and informed view about Britain's position in the World, the view of Britain abroad and the balance of power between the rest of the EU and the UK.

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.