"what is the purpose of the cover over the p/u? Strictly aesthetics?"
No. Leo intended the pickup cover as a handrest, and therefore also installed the treble-side tug-bar. We were supposed to rest the right hand palm on the chrome cover, our fingers gripping the tug-bar, and picking with the thumb. That's the way Leo thought it would work.
The bridge cover had the foam mute under it. So it was there to hold the mute in place. In the old days, with the first P bass (the plank that looked like a Tele), the bridge cover had an open back, and some players installed it "backwards" when they didn't want the mute touching the strings, and flipped it back the "right" way when they wanted the mute.
I play with both covers on my 72 P-bass, and only the pickup cover on my 76 P-bass, took the bridge cover off to allow palm muting.
On my Ric basses I always keep the handrest on. After 30 years it's hard for me to get used to playing without it.
With or without?
Fenders sound good with the foam mute in place but are restricting to me. When I bought my first Fender in 71 it had both covers which I promptly took off and lost, it was unheard of to leave them on back then, same with Ric basses, covers were seen as an anachronism, things have come full circle I guess.