If it isn't self-appointed, are you saying it is the players twisting the GM's arm to do it?
No.
Because otherwise, they're willingly taking on that job. Nobody elects them. They choose to do it because they want to. How can that be anything other than self-appointment? It's not like strange women lying in ponds distribute swords to do it.
I already said - there's an offer and acceptance. Both sides have choice - I cannot directly appoint myself to be GM of a particular group. I can only say I am willing to play that role. I can choose to offer to run a game, or not. The players can then each choose to take my offer, or not.
There can even be negotiation over details before the offer and acceptance are finalized. And maybe the players can ask me to run a game, but I still then have choice to agree to do so, or not.
Nobody is drafted or dragooned to run or play. There is mutual agreement to a social contract. That contract, and the agreement, may be, and often is, left implicit, and that can leave the group overall open to misunderstanding. A Session Zero can help make these things explicit, rather than leave them implicit.
That everyone at the table must agree to participate is, in fact, a space for major exercise of agency for all parties. That possible negotiation is a space for agency.
And then, in the end, the various burdens are choices.
Sure, but his problem is that there is no degree of agency. Agency is binary...
What? No!
I may be given a choice at a particular point. With that choice, I have agency.
Someone else might reach that same point, and not be offered the choice. At that point, they then lack agency.
Now, stack up all the particular points. We can have person A, who always has choices, Person B, who has some choices, and Person C who has no choices. Clearly, then overall, we have a spectrum of overall agency, made up of all the individual moments we have it.
But then, even at a particular moment - Person A might be able to choose literally anything, Person B might have a constrained list of choices, and Person C might have no choices.
Our problem isn't that there is no degree of agency. It is that we have no method to measure it in a meaningful way.