Well put. After Sicily, British troops in Italy had to be put under US command which was better able to get them to fight, and even that was of limited success. Brooke knew that US support would be essential for continued fighting in Italy -- some of if not the worst terrain for combat in the entire war -- and after the southern objectives were in hand (13 airfields, putting occupied e Europe's oilfields and factories in range of British bombers) there was no further point to fighting in Italy. Churchill was unmatched in his speeches, but was always looking beyond the war, how best to position the UK at as little cost as possible, as it devolved its empire; that he was a sneaky little **** was shown during a wartime "Big Three" conference, when he slipped Stalin a note proposing a percentage division of influence over each of the eastern European nations after liberation (it was pointed out that it would be like trying to keep a woman half-pregnant), and Stalin mock-innocently brought it up in front of FDR, knowing full well that Churchill had never told him about it.
Even if we had gone on Churchill's merry Balkan adventure, we would have been tied down by the mountains, just as in Italy, and the Russians would still have taken Hungary first. Salvaging Greece was the best we could do and Churchill admitted as much in his "naughty" note to Uncle Joe.
That Roosevelt "gave away" Poland and Eastern Europe at Yalta is a myth. He couldn't give away what he didn't have and would never have.
We achieved what was feasible despite Winston, by focusing on driving through France and the Low Countries and occupying the majority of Germany. Stalin wanted the US out of Europe after the war, but he failed. Not only did we establish ourselves but kept Western and Northern Europe part of the Free World.