I look at it this way. If you hire a head coach, he's expected to be responsible for everything. If he's an inexperienced head coach, like Mularkey was, it would be expected that he, or the team management, would bring in strong experienced coordinators (like LeBeau) to help him. They didn't with Clements, where MM went with someone he was familiar with, and we saw how that turned out. Mularkey's influence on th eoffense didn't help as was expected either.
If the new head coach is an experienced head coach, and he had a speciality of offense or defense (defense in the case of Jauron) he can bring in a defensive coordinator and assistants that aren't veryy experienced but guys he knows he can work with. The ex-defensive coordinator in Jauron will likely dictate the defense and the underlings will just carry out his direction.
The offensive coordinator selection is a little shaky for the Bills because Fairchild hasn't done it for long. You'd expect, with the head coach being defensive minded, to have a strong experienced Off. coordinator so you'd feel comfortable with him running the show on that side of the ball.
Generally most teams in the NFL aren't strong on both sides of the ball due to coaching and/or players. If they were, then they're likely playoff and Super Bowl contenders.
As we know, this is where Gregg Williams likely went wrong. He had these supposed great lists of coaches but he brought in an inexperienced Offensive coordinator who failed. Gray was hired as Def Coord. with no experience, but Williams had the experience there. Later Gilbride was brought in but his system and the players didn't match.