I think the biggest problem with separating football out is that the biggest schools would take advantage of that to implement structures through which they benefit themselves the most. That's a real problem with what you and
@HailToPurple describe.
That's the key financial purpose of the conference structure, to ensure that the economic benefits (money) are spread relatively equally across all members for the sports that bring the money into the conference.
Here's the issue: The moment you separate football, the football powers with huge attendance/viewership will create a league of 20-25 of themselves where they earn $150 million per school from TV money. Or maybe they take the 30-35 highest or some number. But how do we get a seat at that table? Why do we? 100+ years of Big Ten affiliation is why we have a seat here. Keeping the Big Ten strong (even if it goes national) has to be the only "national" objective of NU's AD.
Why would the biggest brands keep sharing the football TV money if they can keep a regional home for non-football sports without the conference structure and without sharing the money?
That's why I think any small school in the Big Ten or SEC (or even Big 12) would be hesitant to allow a football split off to happen. Nobody would benefit except Alabama, Ohio State, Georgia, Michigan, Florida, Penn State, etc. They'd be able to cut everyone else out in the sport that brings in most of the money.
I don't think that a school like Northwestern would get a seat at the table in such a world; we'd basically be relegated. Maybe they keep us around in non-football sports for that easy access to a big market like Chicago, but we'd be competing against far richer ADs and our boosters would have no incentive to try to keep up with that with just donations but no revenue stream.