Having been without an agile lead for a couple months this summer, I'm acutely aware of how much they do, because they're often doing things that I was just letting drop on the floor as non-emergency.
If you need an agile lead, there's something seriously messed up in the management of your org.
I did actually complain about him to my own manager. It didn't go well.
To be a full time position generally involves a lot of administrative work or covering multiple teams. Talking to management is kind of a catch all but generally it’s interfacing with the rest of the world outside of clients. Legwork on security clearances, making sure all the boxes on contracts are checked, and or enduring the team has all the hardware software they need etc.
It tends to be more important in less software oriented companies.