Do it. I'm a crap bassist but I run the audio output into a Nord Modular and use pitch analysis to generate MIDI that goes to the drum machine (or whatever), and it's great for coming up with things I wouldn't necessarily do on the keyboard. If you're good at bass you'll get that much more out of it with per-string control.
Eventide makes some great pitch related tools and the PitchFactor stomp box seems to be much more quick and accurate than anything else out there... unfortunately it doesn't have MIDI out, AFAIK.
How quick and accurate is the Nord Modular for pitch analysis?
The advantage of the on-board pickups is of course having one pickup per string, but also that they're designed with a specific signal profile in mind (i.e. how far those pickups will be from the strings) and with the dynamics of that instrument involved (i.e. the difference between picking up a signal in an electric bass vs. a flute).
I use a configuration of modules that implements a (very crude) averaging algorithm to smooth away the tiny pitch quirks from the way I slide or hit the strings, and then a note selector module to confine the output to semitone intervals, for when I want to control a different module or produce a completely different sound.
On the other hand, if I want to play a melody and have it sound bassy to begin with (with the kind of pitch inconsistencies from slapping or pulling the strings) the sky's the limit. It's also fun to set the synth doing its own thing and then use the notes to modulate filters or suchlike.
But MIDI pickup is still the route to go if accuracy really matters to you. Everything I record is done with the assumption that I'm going to edit it later. The 'midi guitar' thing cited in OP would probably do a superior job, but I really don't like the ergonomics of it. Another option is to hit eBay in search of a Casio DG-20, an appallingly ugly MIDI guitar from the 1980s that nevertheless has an outstanding midi implementation, or a Yamaha EZ-AG, which sells for $2-300 and has buttons on the frets but real strings on the 'soundboard' and is designed for students. They also made a MIDI trumpet if you can find it.