My take is the players' representatives (captains, no more than two or three) should be part of the Search Committee's early discussions on traits, etc. desired in the head coach. That doesn't put the inmates in charge of the asylum but gives them the opportunity to share with the Selection Committee members what's important to the team.
Then, since the committee will surely use a search firm to allow themselves to cast blame elsewhere to some degree if the coach turns out to be a dud, they likely won't be interviewing more than the top three candidates the search firm
believes meet the job's requirements and assuredly will take the job if offered. No one should reach this stage if there's any doubt they'll take the job, or the publicity over the way RE left will pale in comparison. So maybe there won't be three..........
I don't agree with those here who believe the players should vote on who to select. How many, if any, 18, 19, 20 and 21 year old kids have an understanding of the complexities of the job and can translate that into selecting the right candidate to manage a program? They know what they see in the locker room and on the sidelines but likely don't know all the other issues a head coach must manage. Plus, the reps will learn a great deal about what will be expected of the head coach during those early discussions and can bring that knowledge back to the team.
There could be a benefit to the reps sitting in on the interviews so they can see firsthand how the candidates respond to the committee's questions but I'm not sure we need to go that far.