I think the not caring about “which side of the ball the next coach comes from” is to address candidates like Vrabel and Rivera, who are both defensive head coaches, however the both of them presumably would know how to handle setting up the offensive side for success due to their history.
To me, Brown will have to look extra special to get this job. The final run of this season would have to be undeniably special and he would need to be the reason for it. There will be many good candidates available soon, staying in house for some stupid continuity s%*# is exactly how we got here.