In my band, we often don't run the instruments into the PA. Sometimes we'll run just the kick into the PA. Instead of mic'ing the kick, I put a piezo pickup on it, and we trigger a kick drum sample we like in an Alesis D-4 drum module. It gives us a consistent sound that works well for us.
Whether or not we run the kick in the PA, we have a small subwoofer, like you might have in a car, with a small 150 watt head, and run the kick in there. We like having the kick easily audible on stage with us. As a bass player, it helps me as well.
Years ago, starting in the early 70's, I did a similar thing, using a weird old Ludwig guitar synth box that had a drum sound that could be triggered. We ran that off the kick and ran it through the second channel of my bass amp. I got very used to bass and kick coming from my amp. I did this through the 80's and 90's although the tone generator for the kick changed as technology did.
So, yes...a bass combo can work very well on a kick.