It comes down to looks, in the end. It’s cosmetic trim, with certain secondary functionality – e.g. doubles as a mute holder or a finger/thumb/hand rest. Shielding might have been a theory, but that’s obviously nonsense. It was an aesthetic. Fender didn’t want bridge saddles or pickups exposed.
Removing them is like running an old car without hubcaps and trim rings, exposing your hubs, lug nuts, rims, and wheel weights. It’s a certain non-factory aesthetic that some people like. I think it looks bad most of the time, and the covers don’t get in my way at all on bass. They often do on guitar, but I only really need to remove them if I’m palm muting.
Commonly used guitar technique combined with common guitar pickup placements demanded that most guitar pickups remained uncovered – but if you notice, all ’50s Fender guitar pickups without external chrome covers had either white or chrome pickup covers on the coils themselves. It wasn’t until the the ‘60s that black plastic pickups were left visible, and then it was only on lower end “student” models.
Every ‘50s guitar they released came with a bridge cover: Esquires, Teles, Duo Sonics, Musicmasters, Strats - and Jaguars and Mustangs in the ‘60s. Those who thought they got in the way simply removed them, but they often didn’t get in the way until more rock oriented techniques became the norm. Most electric bass technique of the time wasn’t really impeded by pickup or bridge covers.
The first models without them were the ‘60s Duo Sonic IIs, followed by the Mustang Basses in the mid ‘60s, and Musicmaster Basses in 1970 – all student models, and also the first models with exposed pickups that weren’t white or chrome. They continued through the ‘70s on the other models.