[cantab]

Originally Posted by Fat Kikuichimonji

Socialized medicine doesn't seem like a plausible explanation, given the apparent lack of taxation in the Pokemon universe.

There could be plenty of economic activity incurring taxation that you don't see. Every time you buy something in a Poke Mart, some of the cost could be VAT. There could be corporation tax on the Poke Marts and Silph Co. There could be income tax for those in waged/salaried jobs.

And remember the player character is a ten-year-old kid. I'd bet there are plenty of real ten-year-old kids who know little or nothing about taxation.

Another possibility is that Pokemon Centres are run by volunteers. It's clearly not a difficult process to heal a Pokemon. There could be energy requirements, but there's no evidence of massive power sources or electrical equipment so I'd say there's a limit to that. The equipment could be expensive, but you never see it broken down so it's probably pretty reliable.

Incidentally, regarding

Typically, trainers will withdraw their Pokemon from a fight before the Pokemon actually faints....Being that fainting is highly stressful for Pokemon, most trainers choose to withdraw their Pokemon before it reaches this point.

That's just completely and utterly wrong. NPCs in 1st Gen almost never switch and Trainer battles cannot be resigned from.
So a major part of your thinking for why Pokemon Centres aren't free is based on something that's the exact opposite of the games.

If anything, the Pokemon league would want to hike up the cost of Pokemon Centers, creating a larger barrier to entry for the Pokemon league challenge.

This assumes Gym Leaders and the Elite Four don't want to battle. This goes against Agatha's view at least, "Pokémon are for fighting!" And without challengers, how would the Elite 4 get stronger?
And the whole E4 and Gym Leaders are the government thing is without basis. I think it far more reasonable that they are what they seem to be - the best players in the world's most popular sport, Pokemon Battling.

As for what the government is - well the only thing you ever see are police, so it could be a minarchism/night watchman state. Or the ten-year-old kid factor could explain the apparent lack of government.