Three things I think are high on Emerys list.
1) Have you ever developed top caliber NFL QBs
2) Have you ever designed an offense, game planned and called plays
3) Do you have any previous HC experience at a pro level
Trestman and Arians are the only two that can say yes to all three and I believe those are some basic qualifications he'd like his guy to have. OCs and DCs and can do a great job at that level but then the "Peter Principle" goes into effect as it did for guys like Marinelli, Martz and in times past Wannstadt.
Hiring a guy who has successfully lead a team as it's HC gives just a little extra measure of security that says that guy can do it. Just like in business some guys make great CEO but not every Sr. VP will make it at that level no matter how good he may have been at running his division.