The GM/HC dynamic is such a critical component of success.
The best case scenario is there is a singular goal and a specific plan to achieve it, agreed by both GM and HC.
They both follow that plan in both their assigned capacities and responsibilities.
These responsibilities must be clearly communicated and understood with as little overlap as possible. the chain of command must be clearly established and understood by the team.
The team answers to the HC. The HC answers to the GM. Their management approach is to be unified. They always have each other's back in public.
If the GM disagrees with the HC, the only one who will know is the HC and visa versa. Disagreements are never public. Management is perceived as a singular entity.
When a rift is discovered and publicized, it's over. The team loses confidence and the authority of one or both is diminished to the point where success is unlikely.
Any HC or GM that publicly chastises the other is just hurting himself in the end, not only for the present but most likely in their futures.
A few former Cowboys HC's could probably tell a story or two but they have the professionalism to refrain....so far.