This is just my perspective. I've been in (increasingly) challenging engineering leadership roles over the past 5 or so years.
When I first started, I felt exactly like you do. That my job was to enable and empower my engineering team to build awesome product and that all the "management" stuff was a huge waste of time.
I received some advice from my boss that I was missing 50% of the job. You're being paid and expected to do the engineering leadership side on top of being the interface with the company for progress and business objectives. Any talented engineer with a modicum of people skills can usually do a halfway decent job of leading a small team to build a product, but it takes some business acumen to understand how your team fits into the bigger operating environment of your business.
That was the skill they expected you to bring to the table, and you not only despised it, you blew up at them in a meeting.
Granted, that company may very well have had a terrible meeting process and been looking at all the wrong KPIs for how things should work. But knowing how to positively influence that process inside the environment and doing so in a political way that preserves culture and existing team dynamics is crucial.
I hope you give that type of role another shot. The world needs better engineering leaders that can help businesses understand the value of a strong engineering culture.