So if you like, most colleges have audited financial statements available online, and sometimes endowment reports as well. You can dig into these and find out how the money flows.
And yes, at a high level, there is likely to be some relationship between per capita endowment size and institutional financial aid, but it is complicated.
Basically, this happens because these institutions have operating budgets they need to balance, and what is sometimes called net tuition or similar (which is tuition and fees net of institutional grants) is usually an important component of operating revenues.
But there are other sources of operating revenues–research grants, sometimes net medical system revenues (so this is medical system revenues net of medical system costs), sometimes net revenues from other operations (publishing, space rental, and on and on).
And then there are gifts. Some gifts are for present use, and some are to an endowment fund that then generates usable returns. And some are restricted in use. But even those contribute to the operating budget because if they are paying for something, then you can redirect unrestricted funding to something else.
OK, so the more funding you get from other stuff relative to your operating expenses, the less you need from net tuition, which means the more generous you can be with institutional grants as those reduce your net tuition. And although a lot of other stuff might be contributing operating revenues, endowment returns can be a part of that.
But of course this is also a matter of choice. Like, you could reduce your operating expenses, and then be even more generous with your aid because you would need less net tuition. But that will have other consequences. Like, typically a very large component of operating expenses are faculty compensation. And you may not want to spend less on faculty compensation.
So basically, an operating budget ends up being a choice about how to best balance various competing institutional priorities. And while relatively large endowment returns are a resource that will give you more choices, your choice is likely not going to be to simply maximize aid and minimize net tuition.
All this is interesting but as other posters like to emphasize, it isn’t very actionable from a prospective student’s perspective. Like, presumably you really want to know what this institution will cost YOU. And for that you need to use their NPCs, maybe look at some information about merit policies, and sometimes just apply and see what they actually offer you.
And conversely, you don’t necessarily need to know why they have the need and merit policies that they have, as long as you know they will work for you. But if you are interested, you can look it up.