As I said, the vocabulary and imagination may not be there, but it doesn't mean the distinction isn't made. A lot of England has never really had to reflect on the relationship between Britain and England, because Britain-but-not-England was always something that happened to other people, and it's become clear in the last few decades that the distinction between the two is confusing and distressing to a lot of people, and a lot of them are responding, as the Americans and the Australians did once upon a time, by abandoning the whole project- if not, in this case, the associated territorial claims.
I mean, you note yourself that Englishness isn't trumpted in the same way as Scottishness or Welshness, but the proportion of people in England identify as English-only is approximately the same as those in Scotland identifying as Scottish-only or those in Wales identifying as Welsh-only, and I'd be willing to bet there's no small overlap between that 60% who don't fancy themselves British and the 52% who didn't fancy themselves European.