They were trying to solve the problem by increasing the supply of doctors into the system. The other proposed solution was taking the approach of reducing the supply of money into the system. Neither are contradictory to "maintaining an equilibrium of supply versus demand" but both are trying to move where that equilibrium is by adjusting the supply of two different things. So however you define "while maintaining an equilibrium of supply versus demand" either they were both doing it, or both weren't.
>then you have shortages and wait lines as hospitals go bankrupt and shut down, which will hit rural areas particularly hard
Sure, that is what happens if you only fix the prices in certain states and not others, which incentivizes the doctors to move from states that have such price ceilings into states that don't. That is why the only possible way to implement that solution is nation-wide.
>If you want to lower prices and make sure that everyone gets served
I don't see why lowering the price nation-wide would reduce the supply of doctors. What are the doctors going to do? Migrate? But no other large economy has doctor compensation as high as the US. So as a country, you are only competing against yourselves.
You can always lower prices by fiat proclamation, but then you have shortages and wait lines as hospitals go bankrupt and shut down, which will hit rural areas particularly hard.
If you want to lower prices and make sure that everyone gets served, you need to be a bit more sophisticated than just cutting hospital revenue in half and washing your hands of the consequences.