Are there any benefits for a researcher or university in terms of number of grants submitted?

There are no direct benefits if the applications are not accepted. There might be some indirect benefits, if during the writing process the researcher gains feedback on their ideas and plans from colleagues or reviewers. Or, it might be useful to show evidence to your boss that, in the absence of grant income, you are indeed attempting to secure some.

But one important benefit is mentioned in your question already:

in fact, one of her grants was accepted

You've got to be in it to win it. There is definitely some unpredictability in the grant application process, in terms of who reviews it, who is on the panel, what mood the panel are in on that particular day, etc. So, (all else being equal) the more you submit, the more chance you have.

Importantly, there are also potential disadvantages of submitting large numbers of low quality grants. For example, NERC UK has instigated a "demand management" system for its standard grants, such that if an institution's success rate drops below 20% they will be limited in the number of applications they can submit the next year. The NSF (which is tagged in OP's question) places limits on the number of applications from an institution for a number of its grant schemes. For an individual researcher, this means that they probably need to get through an internal review before they can submit a grant application. In both internal and external reviews, it may not go down well with your colleagues/peers if you regularly submit low-standard grant applications, resulting in reputational damage.

The agencies I have applied to ask for reviews, and the applicant gets these reviews. Those can help to improve the grant proposal, to be submitted to another agency.

I would be hesitant to recommend such a "shotgun approach", as the reviews in the agencies I have applied for are not blind. They cannot be, because the reviewers are also asked to rate the applicant. With a shotgun approach I would very quickly get a bad reputation.

Some universities are not receiving many grants because they are not applying for very many. In one case I saw "submit more grant applications" as a strategic goal. In this case submitting many mediocre applications may help a faculty member get promoted.

Some universities reward faculty whose external grant applications are unsuccessful with internal research funding. There is some quality standard.