I don't think David Byrne & Co ever aspired to sound fat, they wanted to sound sparse and geeky, they were not about to emulate Foghat or Grand Funk Railroad soundwise. Tina's "bass" sound - more a baritone guitar - does the job on that number I think.
I can understand what Ilan - a Rickenbacker man, mind you, from across the universe - feels when he hears Jack Bruce' sixties and early seventies sound, I used to feel much the same. I also used to think Jack Bruce's bending-infested fretless playing horrible, but in recent years I've grown to appreciate the musicality in it (even if its not always what your ear expects). I like a lot of untypical or radical bass sounds for their idiosyncracy, but that doesn't make them sounds I would want to have myself. Likewise I appreciate how Chris Squire and Geddy Lee sound, but it's not my cup of tea for myself.
One bass sound I really dislike is Lemmy's (and I just saw him in concert). It is neither here nor there for me. All that distortion makes it lose bass frequencies, it doesn't really have a full sound and though Lemmy's is power chording all the time, you can - unlike on a baritone guitar which he is trying to emulate - never make out quite what he is playing. Much of the time it sounds like a rhythm guitar played through a broken octave divider that has issues transforming the signal. It's not the style of his playing that I dislike - listen to Space Ritual of Hawkwind to hear how melodic and even lyrical his style of flowing rhythm bass can be, but then he had a largely undistorted sound on that -, I think he frankly has a bad sound. His preferred type of Marshall's lack of headroom might be more to blame for this than his signature Ric or are those pups the culprits?
Uwe