I'm an American of British extraction, 400 years removed. Additionally, we just got back from a visit (mostly to Essex and Cambridgeshire, to investigate ancestor leads). I accept all your explanations, yet I confess I don't understand them all.
We've been watching BBC shows as of late, and the two cultures seem quite similar. It seems there is a kind of international homogenization going on (things more similar than my first visit 30 yrs ago, perhaps driven by media and international business franchises?). Mostly, both cultures have the lefty/righty dynamic, where left is trying to pursue higher ed, assimilate non whites and LGBT, and clean environment (for better or worse) and carries endless shame for past Colonial deeds, while Right supposedly upholds business rights/family values yet has strayed into isolationist Nationalism (and some form of racism), becoming the party of the really rich and uneducated white workers (and carries no shame). Both Countries regularly talk about their impending demise, yet both had made remarkable contributions to the world in spite of their flaws (Capitalism, Democracy, and much of what we love as literature, humor and pop music among them...plus even integration actually) and seem to be guilt-tripping themselves into solutions. Here, you are an African American, Mexican American, etc but NEVER do you proclaim to be British American...our identity has melted into the European American masses. Both Countries are a remarkable hedge against fascist totalitarianism (so far), genocide, human rights abuse ala NATO and the Commonwealth. Re class, it also seems similar, you have Royalty and we have the Kennedys, Bushes, Roosevelts etc. You have Oxford and Cambridge, we have Ivy League (and basically the West Coast equivalents of UC and Stanford etc). Both Countries seem to offer modest upward mobility (including military mechanisms). Your medical more broad and fair, ours probably more selective and pricey (but better if you have money and you're REALLY sick). Both also seem the front lines for a kind of liberal decay (truly a kind of undermining of family structure beget by urbanization, many choices, rise of feminism/LGBT which also has unified the righties) Here, we're almost being polluted by college education, as increasingly nobody knows how to work or fix anything unless they're foreign born. You're right that here, people only vaguely associate you with class, into simple low/middle/upper ranks. Everywhere one goes in the world, vestiges of our 2 cultures are seen, and it is love/hate. Personally, I don't always resent the super Capitalists here as many like Bill Gates and Jeff Bezos become philanthropists (you have the similar Richard Branson, but is this less so in the UK?) It remains to be seen if the waste products of Capitalism/Democracy can be rectified before we tear ourselves in half, perhaps by some new ethic/policy that really does create a true meritocracy (my vote is for a mandatory national service and free 2 yrs of junior college).