For those whose parents came here in the 60s - wouldn't they have some kind of residency permit, in that case, given that Britain did not join the EU until 1973?
Remember that immigration rules, particularly those pertaining to spouses of UK citizens, have hardened considerably over recent decades. I would guess that many EU citizens who have been in the UK for decades arrived as the partner of a UK citizen, or as a child of such a partnership. Citizenship used to pass from the father, so a forty-something born to a British woman and European man might not have UK citizenship despite having spent a lifetime here.
Immigration rules change all the time, and even within the EU they have changed in recent years. Before the early 2000s it used to be more difficult to move between EU countries (we needed residence permits when we moved to France in 1998, for example).