I do pay attention to what the original bass tone of the song and within some reason I do try to do that too. I just try to keep it simple, once we sound check I do not touch my amp at all, any variation in tone is going to come from using/not using a pick or my fingers, or backing one of the pickups off to favor the other one. I.E. Billy Idol "White Wedding" is all with a pick and the neck pickup, Duran Duran "Hungry like the Wolf" All fingers and the neck pickup backed off slightly so as to favor the bridge and get that tight, slightly nasal JT tone.
Does anyone notice? Most of the time I don't think so, my bandmates don't, with the possible exception of our drummer.
It is like all those bad memes that you see, nobody even kind of understands what I do, and even when I stop they still don't really get it.