A bumper is a muffler and protector.
As far as Rambow cues, people forget that he was using a plastic that didn't shatter, was not all that hard and didn't have any finish on the bottom. The butt caps would actually get little dents in the bottoms.
Rugs came around a lot later that Rambow. The typical floor back then was wood and when you wanted to tell someone they made a good shot, you would tap the butt of your cue on the floor. It wasn't until the 50's and after that the average player started to use the increasingly more available and fancy custom cues, ala Paradise, Palmer, etc that a need for protecting the butt cap arose. Palmers for years had no bumpers. It wasn't until the 70's that they bacame standard.
The amount of vibration damping and noise absorption they produce is a function of a niuber of factors including bumnper size and attachment method. As far as affecting the way the cue plays, the bumper is very unlikely to do anything other than mess with the mind of the too easily distracted shooter. If he is paying attention to his bumper he should stop betting.