Not many movies in hospitals

By way of answers, I submit an anecdote that may provide a hint.

In late 1933, director Frank Capra had a script in which he very much believed, titled Night Bus. Columbia exec Harry Cohn was dead set against it. But when a big male star from a different studio became available for loan-out, and an equally big female star from another agreed on the condition of an accelerated production schedule so her holiday plans wouldn't be disrupted, Cohn grudgingly relented in spite of his boilerplate reasoning: "Nobody wants bus pictures. They always lose money." (I note in passing that the film was retitled just before its Feb '34 release: It Happened One Night.)

It's only a guess, but it may have been as simple as that: "conventional wisdom" among producers and execs that hospital pictures weren't box office. Of course, that wouldn't stop a going concern like MGM from occasionally dipping a toe in what might have been considered a risky genre. That same male star's very next film back at his home studio was 1934's Men In White, which took place almost entirely within the confines of a large metropolitan hospital. And a studio with the prolific annual output of MGM could afford a certain percentage of gambles each year, as well as collections of modestly-priced B features like the Dr. Kildare series. At other studios, hospital-centric pictures were more often than not equally budget-conscious Bs, like Warners' 1936 The Murder of Dr. Harrigan and 1938's The Patient in Room 18 (which were, like phantomparticle 's submission, Green For Danger, hospital-centered whodunits).

While there continued to be occasional big-screen visitations, like 1956's efficient Emergency Hospital and 1961's quite good The Young Doctors, perhaps the answer to your second question is that the intimacy of such human dramas was felt to be more suited to the equally intimate, small-screen home-viewing experience. Still, if memory serves, hospital TV dramas had lost their popularity by the close of the '60s other than for daytime drama, and didn't become trendy again in prime-time until the 1980s/'90s.