I think the NIH should more frequently use the power of the purse to change the behavior of Universities. I expressed this recently in the context of a Congressional demand for information from the NIH Director on the NIH oversight of the civil rights obligations of their awardee institutions. I have probably expressed this in other contexts as well. Before the invention of the K99/R00 one saw handwringing from the NIH about how Universities wouldn't hire less experienced PhDs and this was the RealProblem accounting for the time-to-first-R01 stat. My observation at the time was that if the NIH was serious they could just ask Universities for their hiring stats and tell ones that didn't hire enough young faculty that they were going to go to the back of the line for any special consideration awards.
This could also apply to occasionally bruited NIH concerns about women, underrepresented groups and other classes of folks not typically treated well by Universities. Exhibit lower than average hiring or promoting of women or URM professors? You go to the back of the special consideration line, sorry.
My suggestions are typically met with "we can't" when I am talking to various NIH Program types and various grades of "they can't" when talking to extramural folks about it.
Of course the NIH can.
They already do.
One very specific case of this is the K99/R00 award when it comes time for administrative review of the R00 phase hiring package. If the NIH finds the proposed hiring package to be deficient they can refuse to award the R00. I have no idea how many times this has been invoked. I have no idea how many times an initial offer of a University has been revised upwards because NIH program balked at the initial offer. But I am confident it has happened at least once. And it is certainly described extensively as a privilege the NIH reserves to itself.
A more general case is the negotiation of award under unusual circumstances. The NIH allows exemptions from the apparent rules all the time. (I say "apparent" because of course NIH operates within the rules at all times. There are just many rules and interpretations of them, I suspect.) They can, and do, refuse to make awards when an original PI is unavailable and the Uni wants to substitute someone else. They cut budgets and funded years. They can insist that other personnel are added to the project before they will fund it. They will pick up some but not other awards with end of year funds based on the overhead rate.
These things have a manipulating effect on awardee institutions. It can force them to make very specific and in some cases costly (startup packages, equipment, space) changes from what they would otherwise have done.
This is NIH using the power of the purse to force awardee institutions to do things. They have this power.
So the only question is whether they choose to use it, for any particular goal that they claim to be in favor of achieving.