It is possible that good pickups can replicate any tone, I think. My Alembic Orion 4 can sound like a Jazz or Precision and especially an Alembic! I had a Jazz new in 1968 which did me well and then was stolen - that said it needed a bit of wire to stop and earth the buzz from pickups. Also, don't you find, you can use your fingers and position of fingerstyle to get the sound you want? At the end of the day, if you play with other musicians in bands and aren't a soloist then all you need is a good tone - the subtleties of Precision/Jazz/Musicman etc just aren't as important as making music, in my opinion.