It'd be very hard for the BOT to trust the image of the Church (that's really what it is) to a non-LDS coach unless they were VERY confident that this person supported their mission. I could see them maybe going along with Grimes or Lamb who have been here for an extended period of time as assistants.
But even then ... not if there were LDS coaches just as competent. They'll pick the competent LDS coaches first.