They already had much more efficient ICBM programs though, the results of which were also problic. The military interest in NASA was about putting humans in space, They were worried that manned orbital platforms/bombers were going to be the new ICBMs and the US would lose out if it didn't develop them first. Which was a legitimate concern, but turned out to be an incorrect one. The moon shot specifically was a kennedy vanity project with a lot of institutional momentum behind it, especially after he died. America didn't need another black pill right then, so they couldn't just axe it. It either had to succeed or appear to succeed. I think the latter is what happened, but we'll never know until someone goes to the eagle's nest and looks around for footprints. The black budget wasn't the purpose of the program, just an added incentive for everyone involved to stay quiet once the decision was made to fake it.