Governments could issue bounties for drugs that treat diseases like cancer, heart disease, diabetes, etc. Rewards would be based on effectiveness, side effects, how many people are effected, and how severe the illness is. Patents go into the open domain. And while a lot of money might flow to new treatments for big killers like heart disease early on, in later years, other “orphan diseases” would get attention once the higher priorities are better dealt with. Publishing failed research could be incentivized too.
Is there something I’m missing here? Has any nation considered doing this?
A bounty only works when the value of the bounty exceeds the sum of all other opportunity costs. You should probably ask yourself why most doctors or pharmacologists are interested in the practice of medicine and aren't particularly interested in spending their professional years building portfolios of research on rare and difficult to document diseases. Hint: Despite what you claim, there's still a market involved in allocating value to certain specializations and projects over others even in an ostensibly technocratic discipline like modern medicine. You may call it triage, but it's economics at the lowest level.
The medical researchers that do spend their years toiling in obscurity either have enough grant money to do their ivory tower medical research (relatively) undisturbed or they receive so little grant money that they eventually drift into private industry. The person you're looking for is someone who is rational and intelligent enough to do bleeding-edge medicine and yet desperate enough to have one's worth as a doctor/researcher decided by the whim of the government. That's a very small portion of trained doctors and/or medical researchers and most of those would already work at the NIH, FDA, DoH, etc.
It's likely that a bounty won't change things.