As to whether or not the football team does more good or bad for the church's image is up to discussion. Clearly, in the case of BYU-idaho the church leadership, under inspiration, felt that the athletics program took away from education and the image of the church. They chose to discontinue that athletics program. It is almost certain the leadership of the church is considering if the athletics program at BYU-provo is a benefit for the church. Clearly, this was shown when selecting mendenhall for the position. It is accepted that his football capabilities was not the first, or second, or third reason why he was selected.
Furthermore, other universities with outstanding athletics programs have de-emphasized their athletic departments because it was felt by leadership that the core mission of the university was being undermined by athletics. The top two football universities by championship number are perfect examples of this (Yale and Princeton). The University of Chicago is another example.
I submit that because of BYU's commitment to academic excellence and their mission of representing the church, they should de-emphasize the athletic department similar to that of Yale, Princeton, and U. of Chicago.