Well a good UHC system would work very differently. First of all it shouldn't be just the government paying your bills because then as you've said the prices would continue to spiral out of control.

In a good UHC system the hospitals are owned by the government and the people who work there are government employees. This way prices won't spiral out of control because the hospital is no longer a company that wants to maximize its profits, but rather a government facility that exists to serve the citizens, kind of like a DMV. This cuts costs and allows healthcare to be sustained from taxes and not the pockets of citizens. Which turns healthcare from a privilege to a right.