Fair enough but let me ask you this.......are you firing MM and maybe looking for a new 2024 starting QB if advance to NFCCG and lose??? This is the nut and bolt of OP's thread.
I would never succumb.to fan pressure, a mistake I think Jerry Jones made with Wade Phillips in more ways than one. I'm glad Shram wasn't influenced by fans with Tom Landry early in his career. I would take into deep consideration, the circumstances.
What circumstances am I referring to?
The first criteria used would be based on how much authority that HC realistically had to control the destiny of the team. It usually isn't equal to the authority we fans might perceive. In Wade's case, he was given, what appeared to be, dual responsibilities. He was also the DC. In reality, he was stripped of all authority related to the offensive side of the ball, with the exception of live game decisions and practice organization.
Jerry Jones and Jason Garrett took on many tasks typically belonging to a HC. In 2010, things came to a head when they were forced to start the season with backup O-linemen due to injury and they decided to feature a pass heavy offense. This indirectly led to Romo's season ending injury and a 1-7 record in the first half of the season. That was when they fired Wade Phillips. I strongly believe that both Jones and Garrett had way more to do with the team's demise.
Romo injury leads me to second set of circumstances when discussing a HC's culpability in a poor season. There were devastating injuries that season. There have been a handful of seasons, not just the Cowboys, in which an improbable amount of injuries become reality. Another example would be the 2020 season.
How much influence do HC's have in selecting players in the draft? I assume it's different for each team. However, that too has to be taken into consideration. Most likely cap management is not his responsibility, an obvious element of success or failure.
So, my long winded answer to your question is......that depends. And it should.