If you're only thinking about the already-there population, no. But many hospitals have patients that travel to get there.

Building a Mayo Clinic hospital in Arizona results in more sick people traveling there for care than otherwise would have.

Seems like there are some natural ways this could cause a loop. Now we hire more doctors and train more new specialists. That burnishes the reputation even more. Now more people want to come!

(One thing to note is that it's generally always a localized increase, not an absolute one. The people living in the new suburbs that you built a new shiny road to still would've lived SOMEWHERE in the country. You didn't increase that country's total demand for transit. You just redirected it into "road transit from this particular suburb" from wherever it would've gone in the absence of building that suburb.)

Thats because roads are not "supply" it is "price". Building a hospital and staffing it should make the cost of healthcare in an area lower because there is more "supply" of healthcare. Which in turn means more people would go get healthcare that couldnt afford it before. The sick people are already there, they just cant afford to get treatment.

Just like the people who want to drive somewhere are already there, but the cost of a clogged commute makes it not worth it for them to spend the time, so they do something else.

No, but it can make them visible and willing to be treated. If people cant afford private health care or are unwilling to travel long distances or wait for a long time on a public health waiting list, they might just put up with being untreated. This is quite common for things like joint replacement or chronic dental issues.

It increases the number of people getting treated for their sickness, who otherwise would not be.