While that may be an element it also comes from the Knights Hospitallers who would set up rest stops for pilgrims. The thing is pilgrims would often get sick and have to be taken care of by the Hospitallers, which also blends into what you're talking about.

That's why "hospitable" isn't anything you expect the average hospital to be.

In swiss german it still is "Spital".

In Switzerland, the word Spital is in use instead of Krankenhaus