Patients also only get sent to House when nobody else can figure it out.

it's a fake drama without any basis in reality.

Yeah, it's obviously not logical or grounded in reality, but it is at least somewhat consistent within the show universe.

I think during the story line, where Cuddy negotiates with the hospital network insurance provider for a new contract, she makes the argument, that their hospial sort of provides a halo effect. Basically the hospital and probably House in particular (but maybe also others like Wilson) are some of the best in their fields, which obviously is great for marketing purposes, if you can say that you have one/the best and well known hospital in your network.

Edit since I thought of another aspect:

Iit regularly comes up that the team writes case reports. So maybe there is also an aspect where they frequently publish in prestigious journals. Especially since it is a teaching hospital adjacent to a university that might also be an important metric. Similar to insurance providers, the university itself might have an interest in having a reputation for excellence.

There's a hospital near me that openly advertises that it has penthouse suites at the top that feel like swanky apartments. I've never asked about their cost.

"Fuck American healthcare" FTFY

Yes, another in universe explanation, that I already commented somewhere else, is that the hospital is geared towards excellence. There's an episode where Cuddy negotiates with an insurance provider for a new contract and one argument is that they provide a halo effect. So the insurance can claim to have the best in their network or get really bad press for losing them.

Wasn’t there also a multi season arc where they made House teach a class and take on interns so that they had other “reasons” to keep his department

How does employing House make the hospital money?

I thought they did get mad about the queues a couple of times.