Finding a prospective molecule is a huge part of the process - this is why computer binding models are such a hot topic
The fact that we paid for discovery, synthetic development, and PoC is a pretty big deal whether or not it ends up being a marketed drug
I know the guy who sold the patent was paid very well (aftet the university cut, of course) ... but who paid the bills to make it possible?
So if this drug you discovered does make it to market, the drug company itself is paying for the vast majority of the costs and all the risk that goes along with spending hundreds of millions and having nothing to show for it. In return, we dangle the carrot of profit in front of them.
You are correct that the public paid for that initial discovery, but the ownership is given to the lab that found it. The other alternative is for the gov't to keep ownership of the patent.
He found something, and a drug company willing to put up the other 99.99% of the money required to try to take the drug to market (even though the odds of that happening are miniscule) paid both the university and the researcher for the rights to use that discovery (and, as I understand it, usually to get a share of the profits should the drug be successful).
So for a nominal tax-based fee (funding research, not bringing drugs to market), private companies get access to research results that a private company may not have even tried and the institutions involved get a funding bump. And maybe some malady cured as a result. Unless you're a serious "screw you I got mine" libertarian, that seems like how tax supported basic research should to work.