I believe the program is important. It’s where you will be earning your degree and from where employers will hire. Nobody goes to college for an “overall” degree.
My D is at a school that is T10 in her engineering discipline and outside USNews’ Top 50 “overall” rankings. Top engineering employers heavily recruit and she already has two very good full-time job offers. I suspect many of the other majors at the school don’t receive the same attention, but that doesn’t affect her.
Yes, “if you need to change” may be a consideration, but a fairly pessimistic consideration we never factored in. You can always transfer to another school if your major doesn’t work out.
Having said that, it depends on the relative programs’ strengths.
One of her HS classmates had an offer from a T10 engineering major/outside T50 overall engineering/barely T100 school and chose the T15 major/T10 engineering/just outside T50 overall program that my D chose. In that case, the gap between T10 and “only” T15 was swamped by the fall off in strengths outside the one program. (Delaware and Purdue, ChemE, those interested).
So yes, the particular schools would help us provide feedback beyond conceptual/philosophical.
(Edited to add - of course, a student’s comfort level and ability to succeed at a specific school, based on visits, discussion, etc., weigh very heavily against any “ranking”)