Normally not the University. NSF has a "Broader Impact" aspect of the grant applications (for as long as I can remember), and the DOE started to require a Promoting Inclusive and Equitable Research (PIER) plan during the Biden administration. Grant reviewers (typically people from the research community) are asked to take these into account for the review of the proposals.

I suspect the father mentioned above means the latter.

I do not know, but could imagine it's possible, that HBCUs might have their own requirements. But normally, universities do not regulate the proposal writing except for financial aspects (salary windows, IDC+fringe rates etc)