The answer may be that Paul had already turned down John Lennon's request to sing the song for the 'Abbey Road' sessions - it would hardly have made for harmonious relationships within the band to refuse a song to your fellow Beatle, then allow a cover version to be recorded. Stories about what went down at the sessions are legendary - mainly fuelled by the camera presence showing The Beatles at work. Truth is, all bands have the sort of petty squabbles that the documentary shows, it's part of the writing and recording process, and The Beatles had their spats like anyone else. Paul's decision may have been based on his desire to sing his own song - a habit he and Lennon had developed in their earliest days. It may be nothing more significant than that.