The British Empire has been a proxy since the Yalta conference. Operation Ajax was not about dethroning a 'friendly' government, it was about deposing leadership that was willing to suffer economic ruin in order to sell oil on its own terms. Result: oil got sold on American terms.
But that's my point - the British had a huge stake in Iranian oil. They asked the CIA to get involved precisely because they had such a huge stake. The British planned the coup. It was their idea. We certainly had a lot to gain, and we went along with the ride - that doesn't scream to me "VASSAL STATE" Especially when the British had a great deal to gain themselves. Furthermore it is pretty important to note that Churchill asked the American government to get involved on the British government's behest - the Truman administration had been incredibly reluctant to have anything to do with a coup and even Eisenhower wasn't entirely keen on it to begin with. Our relations with the British, though, were deemed more important that Mossadegh's government.
We pressured the British to enter Vietnam. Wilson was adamantly against that. We asked them not to join the organization China created. They went against that. They have consistently taken a harsher stance against Israel than we have. I live in the United Kingdom. I can tell you that the sentiments about American politics, even amongst the political elites, do not seem to point towards anything resembling a vassal or a proxy. Indeed, amongst many of the politicians here during the last election, that was a great deal of derision for many American policies, both domestic and foreign.
Respectfully: I disagree with your interpretation of the historiography. And I must stress that this is all this discussion (which, don't misunderstand me, I feel is quite interesting) is centered around. I agree with the idea that we dismantled the British Empire for our own benefit but I think you are being needlessly dismissive of an entire country's sovereignty.
If you were to simply say we have an unequal friendship, I'd certainly agree. The power-dynamic favors us heavily.