Universities and other publicly-funded research institutions perform basic research for the most part. We study disease X, and find targets Y and Z that might be exploited for drug discovery by pharmaceutical companies. Starting about 10 years ago, a few of the largest universities started small high-throughput screening labs where university researchers could learn how to do some of the screening work done in the private sector. I've used one such facility at the University of Wisconsin. However the number of compounds available for screening is at least an order of magnitude smaller than what is available to pharmaceutical companies. More important than the number of compounds available for testing is the funds to do it. Pharma has it, academia doesn't.
Even in the rare cases where the universities do come up with something that shows efficacy, that doesn't necessarily make it a good compound. How well does it inhibit? How easy is it to make? How stable is it? How soluble is it? What else does the compound inhibit? Does it show undesirable side effects? How easy is it to formulate? Modification of lead compounds is pretty much the middle third of drug discovery, and can take three years or more. Some universities have medicinal chemists, but at least in my experience they do not have time, interest (running their own labs they've naturally got their own projects and interests), personnel, or funds to take a compound discovered at a different lab and work it over for a few years. Even if they did, no university has the funds to get a compound through clinical trials.
Biotech and pharmaceuticals companies fund all the phase I, II, and III clinical trials. The amount of time and money spent on compound modification by biotech and pharma dwarfs the universities, to put it mildly. The same is true for the time and money spent on screening chemical libraries and preclinical trials. Biotech and pharma also have involvement in the very earliest stage of target identification and verification. Universities are important in the drug discovery process. They provide some of the targets, do the vast majority of the basic research needed before you can even tease out a target, and provide all of the early (BS/MS/PhD) and most of the middle (postdoctoral) training of the personnel that biotech and pharma need. But for the majority of drug or drug candidate the bulk of the time and money spent is from biotech and pharmaceuticals companies, not the universities.