Research pays for itself with grants. That's not a conflict
Sports teams in theory pay for themselves with revenue. That's a matter of accounting and debate, but in theory it's just recreation. It's not really a conflict.
Marketable skills vs "how to think" is basically the domain of knowledge you study. Want more skills? pick that major. These can all coexist in the same institution and have for 100s? 1000s? of years.
The issue is the financing, facilities bloat, marketing on the basis of non-academics, inabiility to properly measure academic value, class paranoia, and the rise of the vampire administration class has blown out the cost of a high-reputation degree by a factor of 10.
Universities used to do its core activities/competencies with much less opulent housing, unnecessary buildings, basic auditoriums and rooms, basic labs. I'll put aside research here since grants will fund the facilities they need to do the research. Sports teams should theoretically self-fund their own facilities.
The issue is the bloated cost vs the value of the degree.
What the country needs is high-reputation high-value degrees from very modest/focused institutions. Like military-style boot camps (in terms of austerity, not authoritarianism) where you are focused on the mission of gaining knowledge and not about living in a cool dorm room, watching a good football team, drinking to oblivion. Those are ancillary, they can be there, but they can't be the focus.
That comes down to the institution mission, and selecting students in the right way.
Colleges used to be based on the quality of the people that come out of the institution. Now they are too broadly graded on non-academic and non-performance.