I’m not sure I understand the gist of what you are saying. Of course, Churchill’s correspondence, of course, backs up his point of view. But there is still 2500 years of military history going against that.
The problem with positing a British imperial motive for the Italy invasion is simply that it is entirely inconsistent with Churchill's approach to the rest of the war, especially what he allowed to happen at Yalta. I don't blame Roosevelt quite so much for the latter as he was a dying man at that point, but Churchill essentially ceded the Balkans to Stalin. And Britain, quite honestly, had no imperial ambitions in Italy.