I once bougth a reissue Epiphone Rivoli. It had a good but not very well hidden neckbreak and it was dead cheap. Still got it, with a Darkstar in the neck position and a Duesenberg humbucker at the bridge.
When I bought it I plugged it in and I thought: yes, this is the sound I expected. I started looking at the mudbucker and I saw something strange under the holes. I removed the chrome cover and underneath it was a Seymour Duncan Jazz plckup, with the sides (plastic only) sawn off, to fit! because of the short scale, mahogany neck, set neck and body it still sounded as I expected! In the end a pickup pickups a sound and of course a good pickup makes a bass sound better, but the type of sound/character is really the bass.
I once had a bad, thin sounding USA Jazz Bass and even with a humbucker mounted under the chrome Jazz pickup cover, it sounded thin.
My reissue Mustang sounded just good and with a Nordstrand it sounds fuller and bigger and more as I hoped/wanted.