Why would you think that having say a second pickup and an onboard eq translates to not having a "personality"? I sure don't think that...Foderas, Laklands, G&Ls, EBMMs, Ritters, Alembics, Wals, etc. all have personality to me. That's not to say I don't like Fenders, Rics, and Gibsons (well Tbirds anyway). Basses, like bassplayers can have a "sound" or a "personality". I have no experience with modelers so I can't comment on that.
Extra pickups/active eq/pickup switching does not equal bad tone or lack of personality...why are some people making that leap? You can have a P bass that sounds bad just like you can have a bass with tons of bells and whistles that sounds bad.
Marshall