The schools where a low EFC can be an issue for admissions are the need aware schools. They don’t tend to be the very most selective schools, though, yes they are almost all selective private schools.
Most schools are not need aware in admissions. Most don’t meet full need either. I think you’ll see that the state schools on your list are not going to be piling on the money because your student has high need. They simply gap
Your student will likely get PELL money, subsidized loans and maybe work study. It would be great to get a need grant from the schools as well, but doubtful. As amply said here,getting institutional need money as an OOSer is difficult.