They have gotten involved in lightening the regulatory burden for certain drugs that are orphaned or for rare diseases, which is essentially a subsidy.

Also, the number of drug company booths at a recent infectious disease conference I went to suggests this whole "No interest in drugs that cure in three weeks" thing is...well...not actually true. These are incredibly common infections, there's plenty of market for them. Hell, the compound was initially discovered by Bayer.

I wondered about this and looked it up (leading to my other comment in this thread). NIH has a $30 billion medical research budget. DARPA is a couple of billion. Of course the $30 billion is not all directed at less profitable drugs.

Apparently there are a bunch of hard problems there.

The government is involved, but through academia. For example in Canada we have the CIHR (Canadian Institutes of Health Research) which would fund develop of drug develop (among many other health research).