The ‘63 profile and the tall narrow frets of the American Pro’s make them feel like they’re playing all by themselves, they’re so easy to play. I don’t see Fender trying to copy the late 70’s-early 80’s necks anytime soon. The first edition of the V-mod pickups get a lot of poor reviews, but I like mine so far - and a set of SPB-1’s (my favourite P pickup ever) is really inexpensive to put in anyway. The new V-mod II’s are apparently much better than the I’s. Combine that with a bridge that’s at least two generations of evolution past what’s in your ‘81, lightweight tuners that are about four-five generations past what you have, rounded fretboard edges, a satin finish on the back of the neck, and QC like no other time in Fender’s history makes it a no brainer to me. Fender sold more basses and guitars in 2020 than all other manufacturers in any other year in history.
I get the sentimental attachment - I still have my first real bass, a USA-made Peavey from 1985, bought new. I don’t play it anymore. It’s like night and day, comparing it with a new American Fender. If you are really stuck, keep the ‘81, but if you can swing also getting a new one, try to find a 2008-current one. The worst you face is a pickup swap, and that’s entirely personal preference. The truth is that there are a lot more options for pickups now than there were in ‘81, so I think people are more fussy now than they used to be, and some of us older folks romanticize the original pickups in a bass we’ve owned for 40 years.
Plus a brand new one dated 2020 or 2021 will also say “Corona, CA” on it. Kinda commemorative!