Full pay is a big plus as most of the boarding schools operate somewhere within a 60/40 - 70/30 FP/FA ratio. Andover is the only school that continues to say it is need blind, but that refers to the application evaluation process; it still has an FA budget and maintains a relatively stable FP/FA ratio year after year. Some schools (like Choate) promise to match 100% of demonstrated need for every student admitted. The key is to be a student the school wants (and stats are only one component of a much broader picture).
Also, don’t evaluate schools by endowment. The size of the endowment is meaningless if your student is not a good fit and is not offered admission. If you search the archives here, you will see that even schools with smaller endowments can be very generous with aid for the students they want.
So, don’t let the FA tail wag the BS dog. Do your research to find the schools that are a good match for your student and family values and apply to those. You miss 100% of the shots you don’t take. Don’t let FA need keep you from taking those shots.
Good luck.