So after this year's ceremony for the award of the Presidential Medal Of Freedom to Bill Gates, Grace Hopper et al, I looked up the past recipients. I found to my surprise that modern Presidents after JFK all seem to have been awarded it.
https://en.wikipedia.org/wiki/List_of_Presidential_Medal_of_Freedom_recipients#U.S._Presidents
What is the reasoning here? If someone was a President, isn't having the title of President a big enough bit of recognition?
Well, not all: while GWB's absence can be explained by the fact that the only person since him to occupy the office was his successor, Nixon is conspicuously absent, likely owing to his resignation after watergate, and the fact that it's one thing to pardon him, as Ford did, and it's another thing to give him a Medal.
As for your question, I have some theories, but nothing comprehensive or sourced enough to answer to the satisfaction of this subreddit.