I think they don't hold air as well because they are not sealed as tight as a tube is. They help with pinch flats if that is something you get due to running lower presssure for increased traction. On the trails I frequent, it does not seem very advantageous, but some trails it would.
A flat with a tubeless tire, you can boot stick on boot it, repair or patch the inside of the tire, or remove the tubeless valve portion and run a tube.
I keep the the caps on. I figure they just keep dirt out and my pump heads won't possibly get gummed up. I ride in some bad weather sometimes. Some of this may just be old school thinking and a habit that is hard to break, but for now, I'm sticking with them as they are really not doing any harm. I suppose the weight(which is minimal could be a consideration as well as time in a race to remove the cap and screw it back on).