I'd take him as a DC, but no way he would take that demotion. Some guys are good coordinators, but just aren't meant to be head coaches. I think Rivera and Chud fit that mold, although Chud isn't looking great as a play caller this year.
Fired coaches frequently take demotions when they move to other teams. It would be either be that, or he'd be retained. I can't recall a head coach ever just getting demoted within his current organization. Sometimes coaches lower in the hierarchy are retained or promoted, but head coaches never get to stay if they lose the top job.