Comment by cogman10
Ideally, we'd sever and isolate the NIH (and probably the CDC) from the government and make it operate a bit more like the federal reserve or USPS. We attempted to do that with the CFPB, but having an executive that can simply fire everyone running an agency really messes with the ability for such agencies to properly operate.
Funding would be tricky. Nobody has pocketbooks big enough to send to the NIH other than the US government and we'd run the risk of it going years without money with hostile administrations. You might be able to self-fund if you included someone like the FDA in the mix and charged approval fees. But, ideally these organizations would be funded through general taxation as everyone benefits from their output. Funds shouldn't have to solely come from pharmaceuticals.
Perhaps their funding could be self-generated if they were to patent their findings and license them in such a way to generate revenue where there's some juice to squeeze from the pharma sales.