The early 80s Ibanez Musician basses are really great--neck-through, ebony fingerboards, heavy bridges, very playable with great sustain. Ugly IMO, but I've had one since 1983. Played the hell out of it for eleven years and have never seen a good reason to get rid of it since. For what they go for used, they remain a reasonable option. The one I have (now de-fretted) has a pre-amp that can be by-passed. My only complaint with the bass is the very noisy single-coils (well, and also that I think it's ugly).
I'm not a big fan of the Police any more, but all three of them were clearly great players. It would be a mistake to confuse playing stripped-down basslines for being a poor player.