You are right in that it will vary from person to person whether or not America should be linked, ipso facto, to what Western Europeans wrought in the world. However, after WWII, the US did inherit and become engaged with a great many of the conflicts, controversies and political conundrums that were previously the purview of the Europeans. Israel-Palestine; Saudi Arabia; the overthrow of Mossadegh in Iran (with British help); alliance with Pakistan, and so forth. (Looking outside the Muslim world for a moment, consider Vietnam. That was a French conflict that suddenly turned into a watershed American war).

I do think that it is probably the case that the Cold War had a lot more to do with this than a desire on our part to become the “next” British or French.