Problem with that theory is there is not that much real cash on hand that one member will have. The NFL only allows so much of the Franchise to be held in debt. You can't just borrow 4 Billion and buy out the rest of the family. One member of the family might be able to stay on as a minority owner but why. They should cash out. With the land purchased for the new stadium it makes the Bears one of the most valuable franchises in the NFL. Should sell for 3-5 Billion, maybe more if a bidding war.
I'm not a tax expert, but it is my understanding that the McCaskey's will need to come up with the cash to pay the 30% estate tax on the franchise when she passes away. I believe the land becoming available was a blessing for the McCaskeys because they can use the debt from purchasing all the land to offset some of the tax penalty. Again, I'm not exactly sure how this all works and I could be butchering this explanation.
They'll also be able to parcel some of the land and resell or lease it for capital or an additional revenue stream to help get the financing required to buy-out any family members that want to cash-out. It just makes it all complicated and I believe it would be best for the McCaskey family to have all this worked out before she passes.
As for the NFL setting limits about how owners can manage their asset - I'm not aware of any of those rules - so I can't speak to that. It sounds unlikely.