When Leo Fender designed his first electric bass he put a chrome cover over the pickup and the bridge for multiple reasons.
A. He thought the cover over the PUP would provide some electrical shielding (it's too open to do any good there, but that was one reason- early Precision had a wire going to the cover even).
B. The original P bass PUP was pretty open with the coils subject to damage, so the chrome cover would protect the PUP.
C. He thought the instrument would be played with the thumb, like the way Wes Montgomery played guitar (his brother Monk was an early user of the Fender bass, and did play it that way too). The tug-bar was under the strings to help with this. You'd wrap your finger tips under the tug-bar, rest your palm on the PUP cover, and the thumb would lay on the strings.
D. Everything that was cool in the '50s had lottsa chrome. Aesthetics- the chrome covered up the kinda primitive/ugly parts of the bass.
E. The bridge cover had a foam mute that knocked down the sustain, and helped the bass emulate the double bass.
When the Precision was changed in 1957, they kept the covers for most of the same reasons, except they'd made the new split PUP covered in plastic to protect it. So, when they made the Jazz bass in 1960, they again kept the covers, mostly for aesthetic reasons. Fender shipped all the 34" scale basses with the covers mounted until about 1982. Why? Because they'd always done it that way!
John