I come from an organ background, but don't do running bass lines. I have always put strings, organ,guitars,piano, or some suitable sound for my left hand. I use it for movement, modulation, or added rhythm.
I have been watching other keyboard players and it seems most just use the left hand to define chords with no sound. I expect that many home players would do this as a rudimentary way of playing, but I see pros doing this also.
In the latter case, I begin to wonder if my style is an overkill. In other words, is more better or not. As a caveat, I have been playing for over fifty years, but that doesn't make it right, but is it necessary.
Bernie
Good question
In general it depends on the style if having strings or organ chords on the left hand adds anything.. or just adds to much..