An orthodox Mormon would see playing on the Sabbath as breaking God's commandment and would likely refrain from playing in the NFL, even if drafted.

Some Mormons would not flinch even a little at the thought of playing organized sports on Sunday.

In reality, it is a religious conviction that is determined at a personal level. Church leaders have repeatedly noted that essential services, such as doctors, police etc., is understood and accepted. They have also counselled against playing sports on Sunday. They encourage the members to participate in activities that would bring them closer to Christ.

However, an individual's decision to play in the NFL would not be looked upon by the church's leadership as a point in need of discipline. For example, Brandon Doman played QB for the San Francisco 49ers and was subsequently called as a Bishop (a leader over 200 - 400 people) in the Church. Eric Weddle plays in the NFL and is currently serving as a young men's leader in the church in the area he lives in. Both are looked up to and respected by many in the Church.

In the end, the answer to your question is that it is a religious conviction held by the Church body as a whole. Therefore, the Church-owned school will also abide by that. However, each person (including those playing in the NFL) should determine what "keeping the Sabbath Day holy" means for them and their relationship to God.