I must say that I personally find absolute statements of this kind, without any information about the student/family personal and financial circumstances, quite suspect. At Notre Dame 48 percent of full-time undergraduate students receive some type of need-based financial aid. This suggests that a noticeable part of ND’s student population are full pay, quite a few of them despite merit aid offers from public and other private universities during the admission process.
The point is that a decision in this respect will always be considered in the context of student interest and specific family circumstances and there is no absolute right or wrong, as the personal opinion expressed in the above posting seems to suggest. In the absence of additional information from the OP, there isn’t much that can be added, beyond the good advice which has been offered already by many folks on this thread.