I think a GM needs to know how to operate with the confinements given to them by those up top. It happens in everyday life, and in all organizations. Your boss wants you to make something happen, but makes it a little bit more difficult than it needs to be due to certain constraints out of your control. So you either make it happen or find an organization that better fits how you need to operate.
With that stated, it's long enough to realize that one of the following is true: Hahn is bad at his job or Hahn sucks at working within the rules his bosses want him to play. Regardless of reason, he's shown enough that he needs to be put through that door.
In my mind, I think he's just flat out bad at his job. I appreciate that he breaks the 4th wall and gets upset like we do and shows it. I was happy enough with him blowing up the team a few years ago and getting some major players like Eloy. But, IMO, whether it was due to payroll, or teams overvaluing their players, or him just not having a plan, it's clear that he doesn't know how to do anything other than blow up a team for the Chicago White Sox.
His forecasting has sucked when it comes to players wanting more money (Machado fiasco), his use of assets made available to him such as trade chips or money has sucked (that bullpen, Leury, etc.), he has allowed his players to become depreciating assets (Grandal, Kimbrel, Hendriks, etc.), his answer to filling starting voids on the team is to have players way out of position or to fulfill with backup minor league options.
IMO, I can't put that on Jerry. If Jerry told him that they were going to roll back on payroll, that's understandable because they sucked in the worst division in baseball. Hahn should've sold off the team at the deadline, especially Abreu. He didn't even do that. So now we are stuck with the exact same team again and expecting to do 12-15 games better based upon hopes and dreams.