Lots of interesting myths about John Wayne.
Regarding whether he paid the government for the Wild Goose; he did not. He bought it from a prior owner who would have been the one to buy it from the government:
Minshall served 16 years on the vessel, where Wayne relaxed and entertained his friends. Only a few were movie stars, such as Claire Trevor, Hugh O'Brian, Dean Martin and Maureen O'Hara. Two of his favorite shipmates were Newport Beach car dealer Chick Iverson and Max Wyman, from whom he bought the boat in 1962.
Wayne bought the boat for a little over $100,000.00
As far as him wanting / trying to get into the service, there's this:
He did well at school both academically and in football. When he narrowly failed admission to Annapolis he went to USC on a football scholarship 1925-7. In a 1976 interview, he said the greatest disappointment of his life was missing an appointment to the U.S. Naval Academy by one position.
And here's a very cogent discussion here:
The Straight Dope: Was John Wayne a draft dodger?
I recall having watched a show, possibly with Lucille Ball, and other old-school Hollywood types, talking about "the old days" and commenting on the situation talked about in the link above, wherein Wayne's career was just taking off and he was not yet solidly established. In fact, arguably, it was during the war years that his carreer really, finally took off.
Angel and the Badman showed the studios he could be a successful leading man, but it was not until the war movies that his career really took off. Bottom line - it was a business decision. Could there have been a little resentment about not getting into the Academy? Maybe, but I doubt it, considering how many of his military roles were naval ones, and he did buy the minesweeper!
My grandpa was 2-A deferred during the war, because he built Caterpillar tractors for the Seabees and Engineers to advance across the Pacific and Europe. He never got a lick of grief about it from his two brothers who fought in the war.