I think the tone you hear is more from the amps that were around at the time, and the recording process rather than the basses themselves. There's really not much difference in an old bass and a new one, except the old one is old. They're still made the same way, though much better now, in the case of Fender, since '08. IMO.
I've played quite a few 60's basses. I can't remember them all, but my best friend growing up had one of the slab P basses. She still has it today. I started playing in the mid 70's so 60's basses were pretty easy to find back then.
Regarding amps and the recording process: That's a good point and probably very true. That's an interesting perspective, especially regarding the relative quality of new versus vintage Fenders. It would make the prices people pay on eBay seem a bit silly.