In those famous White House touch-football games that the press described as involving the Kennedy brothers, others in the clan, and visitors, JFK did not play unless it was for a photo-op. Jackie was injured in one game where JFK was only a “spectator”.
You are correct in that he did officially play football at Harvard.
That link I provided also had a few interesting tidbits on a few other Presidents. Off topic ... but I found the Nixon play story kind of interesting. Gerald Ford was quite a player despite his image portrayed by Chevy Chase. Your comment triggered me to look up that info. Thanks.