...most others due to Honor Code, admissions issues, LDS culture, limited recruiting pool, etc then those same extenuating and complicating factors also affect the AD's ability to discharge the duties of his job effectively as well. Maybe you could even argue it makes his job harder than it does the HC's job.

If the HC at BYU has a more limited recruiting pool than the HC at the typical g5 or p5 school, The AD at BYU has an even more limited pool of potential HC's to pick from for his vacant coaching position. The HC can recruit non-member athletes and often finds total studs in that pool like Curtis Brown, Ronnie Jenkins, Steve Sarkisian, Jim McMahon, Omar Morgan, Tim McTyer, Jenaro Gilford, Jamaal Williams, Jordan Leslie, etc. Some struggle at BYU while others don't, it's definitely a mixed bag. When it comes to finding the next HC however, the AD at BYU can only hire someone approved by the BoT and in viewing all the hires in my lifetime that has required them to be active LDS, current recommend holders.

This severely limited pool applies to all the sports teams at BYU (I'm not aware of any sports teams at BYU having a non-LDS or less-active LDS coach).