All of the Fender bass bridge covers had rubber mutes except for the the first few years of the Jazz Bass when the mutes were mounted to the body. Mutes were an seen as an essential element for electric basses for the first few decades, and many major manufactures (Fender, Gibson, Rickenbacker, Gretsch) included one in their designs.
Aside from the supposed grounding function on the earliest Precision basses, I think the covers were designed to for muting, protection, aesthetics, and to serve as a hand rest when using a pick...a design element carried over from the Fender steel guitars. The covers didn't hinder hand placement as most bassists in the 50's and early 60's plucked or picked the strings close to the neck, as evidenced by the pickguard designs and placement of the finger/thumb rest. Plucking/picking the strings over the pickups or close to the bridge is a "modern" concept.
I don't think the covers themselves affect the sound in any meaningful way, outside of the mute.
- Steve