My D also got called off two WL. One school called on May 3 and offered merit aid, even though she was full pay. For that school, all she did was accept the WL position. She declined the offer. For the second school, she sent emails and stated that she was full pay and would definitely attend if accepted. That call came in the first days of July. We don’t pretend that being full pay didn’t play a part, but I am also sure that her genuine interest was clear. She does not fill any special niche.
Some schools that are need blind, are need aware during the wl period. If a college needs to go to the wl, they are going to check and see what their financial aid budget looks like. If they are over budget based on who has accepted spots thus far, they are likely to prefer kids who do not need aid. If they are under budget, then they may be able to accept kids who need aid.