It's assumed that some trainers you meet along the road even in the games have earned as many badges as you have at that point in the game. The dialogue of some trainers in some games suggests this as some actually admit to having a few badges.

For convenience, they really don't have a lot of challengers, that's because each time they focus on the main character. In the Pokémon Origins anime, it focuses on Red. In the regular anime, it focuses on Ash. In the games, it focuses on you. Of course you wouldn't see other challengers.

But based on what I saw in the Origins anime it looks like they would have some kind of healing machine set up somewhere in the back. It's very structured from what I saw.