I must admit its something thats always puzzled me a fair bit.
Paul always recorded his Bass parts with no sustain, (in other words when he plays a note its very short, not a long lasting note)
I used to think its was inherent in the Hofner Bass he played, that it didnt sustain a note very long, but when he played a Fender or Rickenbacker Bass he did the same. So it was a Bass sound he liked obviously.
Indeed nowhere is it more pronounced than on Yer Blues where he plays his newly acquired Fender Jazz Bass, now I've had a few Fender Basses and I know they are good for sustaining notes but Paul plays very short 'plucky' notes on that song as he always did on his Hofner.
Similarly when he played the Rick on Hello Goodbye, its very plucky.
If you listen to Keith Richards playing a Fender Bass on Yer Blues (Rolling Stones Rock n Roll Circus) his notes are sustained (which I sort of prefer).
Interestingly when John was recording Double Fantasy he employed renowned Bass player Tony Levin, I remember hearing John say to him when he played "ahhhhh...we never had that great Bass sound in the Beatles"............I wonder if John liked the sustain Tony played with ?
The only 2 songs I can think off of the top of my head where Paul used sustain were Come Together and Rain. (there's probably more)