By What process does a Mage gain the rank/title of Archmage? Obviously the way the player character mage got it is far from the norm. Is all you need is to be a powerful enough magic user? Or to know a large repertoire of spells? Is it a politically appointed position? Is leadership and organizational skills more important for this position than magic power? Is it a official title that has authority in of it self, or is it just a title given to powerful mages, and their authority comes from their magic prowess? Can only the Kirin Tor Appoint Archmages, or can other kingdoms appoint them as well? Do Archmages command authority in Dalaran, or only Respect from lower ranked mages?
Who knows.
You’d have to be powerful, and be friends with the council.
Likely, and probs.
This is a question where there isn’t really an answer given in the lore.
Mage is like science in this world, you need to be pretty adept and study a lot the subject, there are many branches in which mages tend to specialize.
According to lore, the title doesn’t necessarily means the mage is extremely powerful, one example we have is Antonidas which was powerful but not much as our current lore figures (Jaina, Khadgar, etc…) however he was a great researcher and published many papers. So it’s safe to assume it could be given to any mage for extraordinary performance in either research or magical display.
Now, is it a political position? An Archmage can be a leader or politician however, the Archmage title doesn’t mean leadership or political power.
The Kirin Tor isn’t the only one with the ability to name Archmages, as we’ve seen Archmages in the mage quarter in Vanilla and back then Dalaran was no more. However they could have come from Dalaran and settled in Stormwind.
Generally though titles differ from places to places, such as Blood Elves having Magisters as the uttermost authority regarding magic, or Nightborne having similar terminology.
they perform magic while standing on an archway
If I’m remembering right, to become an Archmage you need to be Master level at two schools of the magical branches, and be at least adept level in the rest. There are seven schools, with necromancy being the debated eighth, as per the little books all over Wrath era Dalaran detailing them as such. Because of this, Archmage itself isn’t a low bar, but it’s not exactly an impossible to reach one either. Not all Archmages are equal in terms of wealth, influence, or power either, but every Archmage is considered a fairly potent magic user.
Contrast that with Archdruid, where the title seems to only be granted to those who are either the most learned and powerful druid of their race, or the most learned and experienced of a particular totem, and you have an entirely different and more nebulous way of gaining said rank.
The Archmage and Archdruid titles seem to have become largely arbitrary titles to be handed out in the modern day. Come BFA, there’s just three Archmages who serve absolutely no purpose hanging out in the main bar in Boralus, and you can summon an infinite supply of Archdruids in the Darkshore Warfront.
Basically, it just seems like you gotta be really good at what you do, or at least passable.