The term "general contractor" gives the impression that the NIH comes up with research plans. As you might know, this is not the case. The individual researcher (PI, "Principal Investigator") at University XYZ comes up with a hypothesis to solve a potentially interesting problem, designs experiments to confirm or disprove it, and asks the NIH for a grant. In 5-20% of all cases he/she will be successful. Officially, the money goes to the university which then passes it along to the PI. It is primarily the university's duty to obtain the required "financial disclosures" of outside funds from the PI. Believe me, I have to go through this process every year (our university wants the data even if you are not federally funded). Of course, I am not under the illusion that the university can check every detail in my disclosures, especially if foreign funds are involved. This task then falls to the NIH, and I agree with you that for a long time they did not do proper checks. I attribute this to sloppiness, or lack of awareness of the problem, and not to evil intentions. But we might differ on this conclusion.
And when an NIH grant expires, you are not "fired". You still get your salary from the university for all the other things you do (teaching, service).