According to USNA, less than 30% choose the primary method. Some even allow the boards themselves to select the method, based on their findings.
Most, if not all, MOCs appoint a board to do the selection process. A bipartisan board composed of both civilians and retired military personnel, and often including educators. I find it difficult to believe, that with these checks and balances, that politics is a widespread issue. Like you said, maybe isolated. The Academies, of course, like to make the final determination and, thusly, track the 'primaries' very closely.