They don't have an interest in fixing the problem. They want to get as much research done as possible---and so they benefit from exploiting cheap labor. (One could perhaps argue that they are harmed in the long run, but this isn't clear.) They also don't see PhDs leaving academia as a bad thing at all. To the contrary, they advertise it as a positive externality; the research money isn't just staying inside an insular academic community, but is giving companies across the economy access to skilled labor.
Giving money to staff scientists would work against both of these benefits. It would cost more and could reduce the number of trained scientists entering industry. (Although that isn't clear. Creating more NTT positions might just give hope to more of those eternal postdocs.)
> They want to get as much research done as possible
I agree, but it certainly does not feel like the NIH has thought about this carefully either. My impression is that the NIH does what does because this is how it has always been done (and that worked well enough for the PIs they consult). I would love to see data suggesting that the current arrangement is optimal--or even close to it. The only thing that comes to mind is a recent paper arguing that peer review scores for grants were mildly predictive of the number of number of resulting publications.
As for staff scientists, one good staff scientist could certainly be more productive than a gaggle of newbie masters' students on their own. Right now, everyone is incentivized to do just enough to get the next paper. Permanent employees could build up infrastructure (protocols, code, etc.) that is actually reusable and ultimately more efficient.
Giving money to staff scientists would work against both of these benefits. It would cost more and could reduce the number of trained scientists entering industry. (Although that isn't clear. Creating more NTT positions might just give hope to more of those eternal postdocs.)