Just a couple of thoughts here.
It is true that there is a lot of financial aid available and many students receive it. But you should assume that at most schools, it does have an impact on your application and a big one. In short there are two buckets and the buckets are not equal in size. 65% of admissions come out of the smaller bucket. 35% of admissions come out of big bucket. I got the impression that you are choosing to pay tuition currently with f the hopes of getting in, and receiving aid in BS. I’m not sure this is the best strategy, certainly not the one I would pursue.
Our experience with full day and financial aid admissions with the same school and same kid were very different.
Day students are different. But if she wants a boarding school experience and you need financial aid, encourage her to expand her search to include several schools with a higher admit rate.