There is a large chunk of schools known as “tuition dependent” where being full pay is going to help. Help how much? None of them are going to bend over backwards for a kid who is clearly unqualified, but a kid who meets the bar is going to get a second look.
Endowments are important. Research dollars and productivity are important. The role of the grad programs is important (Harvard famously budgets so that each graduate school has its own budget and is financially independent- so the Divinity School does not benefit from the Chan Public Health’s endowment). The amount of non-educational revenue is important (colleges have developed for-profit revenue streams by leasing land, becoming commercial property owners, etc.) The amount of revenue from patents and intellectual property-- huge at places like Stanford for example, where professors have created profitable companies on Stanford’s dime.
All of these are factors. But colleges which balance their books with tuition and don’t have these other robust sources of revenue are going to care about full pay.
But not the Ivy League. They don’t look and they don’t care. They don’t need to look at whether or not you’ve applied for financial aid- even if you are the rare disadvantaged kid growing up in Greenwich CT or Atherton CA or Chappaqua NY-- and they do exist- admitting a few “they need more aid than their zip code suggested” is going to be a plus, not a negative.