It does sound like a good strategy. We gave both my Ds that same advice. They were both NMF and had options of full rides. We gave them a max budget and said if they could match it with scholarships, the school would be considered. We also said if they took a full ride, they could keep half the savings.
You are leaving off a couple of great schools for NMF - University of Alabama and my favorite, UT Dallas. Depending on her major, they should be considered.