That's not necessarily true; fewer limitations on who is allowed to provide health service and manufacture medicine and health technologies would also drop prices. Competition in healthcare is very, very difficult because it's often illegal to compete.

You're actually talking about the same problem. Competition in healthcare is heavily restricted by stuff like certificates of need because there is little cost-sharing with utilizers so you get overutilization & induced demand. This is Roemer's law [0]. A large part of the reason for competition restrictions is because of a hacky attempt to control overutilization without direct cost-sharing.

0: https://en.wikipedia.org/wiki/Roemer%27s_law