DSY – i am somewhat surprised that you got the responses you did. maybe it is my oot naivete but i never heard of a yeshiva saying not to apply without funds. They may tell you to apply and promise funds you dont have and make you sign forms that you agree to raise the funds that you don’t have (happens often), but it is sad if true.
On the other hand, many institutions have closed their doors in the face of boys who are a bit too weak or who need “too much support” because the schools don’t have the funds to support them and they figure that absolves them of trying.
Applications are sent in separately from financial papers. Applications should be accepted and processed and the student should be accepted before the finances are even brought up to an institution. Get paperwork together to verify the boy’s current levels, samples of his work, letters of reference from chavrusas or similar and send in some applications. You will bring up finances when you get your acceptance letter.
I know that minimum tuition here for those who cannot pay is about $3000 a year and our yeshivos don’t charge separately for dorm or food. There are options.