So the answer to your question is why it is so tricky to find a good head coach.
No, you absolutely do not need for your head coach to be an offensive coordinator. But what you DO need for your team to be successful long term is for whomever you bring in to have a SYSTEM.
What I mean is, they may not have the details per se on the interview, but they need to be someone who has or is willing to develop a core identity of what the Bears do on offense. This can and should slightly change season to season as other teams in the NFL catch up to what you are doing, but there should always be a core identity there.
And whatever OC you bring in, is coming to run YOUR system, NOT theirs. Now, if they have some concepts they like to call that fit into YOUR system, you certainly can accomodate that as part of those year to year adjustments you make to the system. But it will never be completely THEIR system. It will be head coach's core philosophies, built on whatever synergy the coach wants with the defense.
The problem is that you don't have a lot of head coaches who are smart enough to want to give a team a total identity. Most Head Coaches only care about their side of the ball they came from, and let the other side have free reign. And because the league has become offensive in nature, teams fear losing good OCs, so they look for the next hot one to be their head coach, which they may not even be ready for.
Its why its so hard to find a good head coach - you have people like Nagy, who want to be OC in practice but still have that shiny HC title. Meanwhile he's woefully prepared to functionally run a team.
If it were me, I wouldn't look at OCs or DCs. I would look at the most successful Special Teams coordinators in the league for Head Coach. Because its the job closest to what a head coach has to deal with, not to mention you have to take the bottom third of the roster and make something good if not great out of that, and because it is the bottom third of the roster, you constantly have to deal with turnover, be it from the GM replacing the bottom parts of it, or because of injuries, some of your best guys are now pressed into service for the offense or defense.
If there are coordinators who are creative enough to stay on top of that well enough to consistently be top5 to top 10 special teams in the league over a period of 5 years or more, or if over a period of 7 years can rebound from a dip out of the top 5-10 and then reclaim that spot and keep it, THAT is someone I would have on the top of my list to be a head coach.
Because with all that change, someone that good at keeping Special Teams afloat will be a very flexible-minded individual, which means you don't run into the hubris you often see with these offensive and defensive "geniuses". And that means as a GM, if you want to develop an identity on the offense, and an overall Chicago Bears identity, a great special teams coach is going to be the most flexible in willing to listen, as well as the most creative in contributing ideas to BOTH sides of the ball....