Part of the agreement between the tribe and the U is that the U will support higher education for tribal members - including, from what I recall, providing a certain number of scholarships for members of the Ute tribe to attend. So that's the answer to your question of why should a university help - because that's part of the name use agreement.

This article apparently says there are fewer Utes attending than in 1951 (why pick that year?), but is that because the U isn't letting them in? Or is it because fewer are applying to attend?