Why do people talk all the time about bass players playing only one note? There is a very real reason. I'll explain.
There is a concept called "low interval limits" - when you try to play an interval (an octave, a fifth, etc) lower and lower, at some point, the result is mud. This is EXTREMELY relevant to playing bass. You can play unison notes anywhere and they'll sound good. You can play octaves anywhere, and they'll sound good. But a fifth, you will find that, in the lowest octave of your bass, they mostly don't sound good - towards the top of that octave, they start to sound OK.
A musical fourth (which is a closer interval), you need to go higher for it to sound OK. A third, even higher. Better gear will make a VERY small difference with this - the best rig on the planet might buy you another half step or two of acceptable sound for a given interval, as it has less intermodulation distortion - the issue with close intervals down low is that your ear is hearing intermod products, most of which it's making up itself; adding some from the gear makes the situation a tiny bit worse. As I said, the gear is a minor part of the equation - the best rig in the world won't get you past the fact that there is a floor under which you can't play a fifth, and have it sound good.
If you're trying to play chords on a bass, you are going to run into this - it is inevitable. Yes, you don't hear the problem unplugged, because an electric bass, when played acoustically, doesn't radiate much in the way of fundamentals - they are low enough that your ears aren't really hearing anything down there - you're just hearing harmonics. Plugged in, the fundamentals are there, and the problem is there. Turning off the bass in your signal will "fix"the issue, as you've figured out, but why would you play a bass and not want to hear bass? All the technique and the best gear in the world will not get you around this - it's something you need to learn how to work with.
We play an instrument that has limits, and this is one of them. In return for this limit, we get to play the thing that shakes the place (and gets people's butts moving). It's worth it.