No, because this often repeated trope simply isnt true. Government funding usually is very early in r&d, (think lots of small grants for tens of thousands of dollars which have 1 in a million chance of succes). private industry takes the outputs and spends about 2 billion dollars per sucesscul drug screening and further developing it. It is like saying a large part of web development is government funded because tc/ip came from a government lab
No, because the part the government funds is the exploration and initial testing of compounds. As someone else pointed out, it takes quite a bit of work to get those compounds from "works in a lab" to "available at CVS". Many compounds work in a lab, or work in animals, but fail in humans for various reasons.
Academia also isn't a great home for large-scale trials, especially long ones.
Most academic "things" (grants, hiring/promotion/graduation criteria) expect people to produce first-author papers, so projects are usually set up to involve 1-3 people over 1-3 years.
A trial, on the other hand, needs lots of people, ideally at many different sites, and will hopefully produce a single, well-defined outcome. Somebody's also got to make the drug--and under conditions where the end product can be given to humans (i.g., GMP). None of this is cheap either, and the NIH budget wouldn't stretch to too many large Phase III trials.