This was my experience in a TLM role - you have to manage down to your ICs but you have little lateral or upward power. You're basically just conveying whatever your manager decides to do with your team, but with all the additional responsibilities of a staff engineer.

In big FAANG-style workplaces, I don't think that middle managers without the TL- prefix have the kind of influence or leverage you're talking about here. It changes at VP level, but ultimately, most of the corporate management hierarchy is just spreadsheet misery.

its called the straw boss.

as in "and the strawboss said well a-bless my soul, you load 16 tons.."