I just so happen to have been going through some pickup experimentation with my current two main basses (a Precision and a single-coil Telecaster style bass). I play mostly flat wound strings now and like my action low because flats with high action make me work too hard plucking the notes.
I've gone through many Precision and Precision/Jazz basses (pickup wise) over the years, and I also did a lot of pickup swapping with a '75 Reissue Jazz bass I had many years ago.
I started my pickup twiddling with the P/Js, because I was always afraid that the stock Fender J pickup was going to give me 60-cycle hum problems. After doing a lot of reading (this was probably 20 years ago), I decided upon the Bartolini 8s/9J pickup set. I was happy with Bartolinis for a long time. (I'm talking over 20 years.) I still do like them a lot, as they give you a very hot signal that results in just a touch of distortion at full volume that's pretty much perfect for rock and blues music.
The other factor is that I was never happy with the sound I got out of that '75 Reissue Jazz bass. I always got too much metallic sound (fret noise) with the stock pickups. And the sound was a bit thin to me. (It took me many years of experimenting and then playing a Precision to realize that I was trying to get the punchy sound of a Precision out of a Jazz bass pickup arrangement which I found to be pretty much impossible.)
Because of these two issues, I just assumed that the stock Fender pickups were going to be insufficient. I've since modified that stance (at least for Ps), but I guess I bought into the hype for a long time.
One thing I recently discovered was that a much bigger change you can make is to adjust the pickup height, relative to the strings. I've found I like to have mine sticking above the pickguard by about eighth of an inch. with my current low action, this places the strings about 3/8 of an inch above the pickup. Some pickups I installed put the default pickup height lower (some even flush with the pickguard). I find this too low, as the pickup doesn't have enough "oomph".
In my current main bass, I started with a Warmoth body and a Fender MIJ Precision neck. When I first got the Warmoth body, I installed a Bartolini 8S pickup with the height set as described above. As I said, I was very happy with this configuration for many years. Only recently have I started experimenting with new pickups because I found the string to string balance lacking a bit (the E and A are perfect, the D and G sound much lower, especially at higher volumes). I don't know if this is more a factor of the amp I'm using, the relative pickup heights across the strings, the neck itself, etc. Since pickup replacement is a relatively inexpensive experiment (I mean even really nice pickups are $150-200, compared to many basses costing $1500 and up on the used market (and I'm talking only about stock production basses like Fenders and Gibsons, etc., let alone more expensive basses like Nash, Lull, MusicMan, Lakland, etc.), I figure why not experiment a bit. With even cheaper good cables going for $40 and up, what's a pickup compared to a few good cables?
The current pickup is a Lollar 10% overwound. I installed just the rubber adhesive spacer provided and then the pickup, and it was flush with the pickguard. (Who knows if the pickup cavity in the Warmoth body is deeper than that in the stock Fender Precision body.) The pickup sounded beautiful, perfectly clean, with nice tight lows, and no apparent distortion at full volume. However, it did not have enough gain (compared to what I got with the Barts). The height thing bugged me though, because I always put the Barts a eight inch above the pickguard, so I took some of the foam from the Bart installation and stuck it back in there to get the Lollars to the same height. Bingo. Plenty of gain, and still no distortion. I suspect that's how I'm going to leave the pickup arrangement on this bass for the foreseeable future. It sounds really nice to me and I can't imagine why I would change it anymore. But I am a tinkerer by nature (or so I've discovered), so who knows?
One other experiment I performed on this particular bass was to try a Fender 60's Reissue Precision pickup. I was curious to check out the "vintage" Fender P sound. I found it sounded a lot like the Lollar, but with a bit less gain and a little bit less "sparkle" if that makes any sense. I thought for the $65 I paid for it (however, I think their price has gone up a fair bit since just a few years ago), that it was a very nice sounding pickup. I found that I really liked the clarity, compared to the Bartolinis.
This is already too long a read, so I'm going to stop here.
One thing I'll say is that we're talking about really subtle differences in sound between the stock Fender Precision pickup and the Bartolini and the Lollar. I think a long-time bassist might notice the difference while sitting onstage, but almost anyone else probably won't be able to tell any difference unless you maybe pointed it out to them. And I suspect no one can tell the difference out in the audience, because it's buried under the rest of the band and the PA and all the other factors. So, keep in mind that you're kind of doing this just for yourself. I'm not saying that's a bad thing. You're the person who has to be happy with your tone. But we're talking minuscule differences that are going to be lost on almost everyone else.
, is that with modern equip (pups, amps, etc) I can get pretty much any tone I need - with a ‘stock’ instrument… but I do notice a difference with pup changes… one can decide for themselves if they’re classified as upgrades - or not 
I have gone (back to) Roto rounds on my black bass & moved PW’s onto my 3TSB!
