Conference (MVC) had no interest in sponsoring football. I did find this article that explains why the Gateway decided to rebrand itself as the Missouri Valley Football Conference (MVFC). I found the statement below to be very interesting particularly since the MVC appears to have no interest in sponsoring football.

“This is going to allow the Missouri Valley Football Conference members to take advantage of year-long positive exposure received by the Missouri Valley Conference,” said Missouri Valley Football Conference commissioner Patty Viverito.

I guess the reason the MVC went along with the plan is because it gets to be affiliated with DI-FCS football (in name only) and it doesn't have to assume any of the cost or the administrative responsibility associated with sponsoring the sport. I suspect most people naturally assume that the MVC and MVFC are the same organization either because both names have "Missouri," "Valley," and "Conference" in them or because they remember that the Big East Conference and Big East Football Conference were essentially the same organization during the BCS days.