Your concept of "every level of understanding" does embody the PM stereotype. From an engineering perspective, most of your list tends to be 90-95% timewasting fluff and toil. Hopefully with as much as 5-10% valuable content that can help the team to accomplish their mission.
A good PM (IMO) would take responsibility for dealing with the fluff, and engage the TL with the 5-10% useful fraction.
Also, a good PM will strive to learn and understand what the team is doing and why... well enough to act as an advocate for the team, and explain to the many people you interact with throughout the organization.
A team will respect and appreciate a PM's effort to meet them on some technical level of understanding. Being embedded with a team of experts in a particular technology gives you a unique opportunity to raise your understanding of their tools, technology, tasks, challenges, and successes.
A good PM is valued as a colleague contributing to the team's success.