Okay, now hold that thought. Now explain why the CoC or the Nerevarine can't learn the same magic spell that the teleporter casts on you to teleport you.
Because it would be completely useless? "Teleport other: teleports one person (not yourself) where they want to go." As far as a hero's needs go, it's completely pointless. You can't use it to travel (since it doesn't teleport yourself) and you can't use it for attack (since it's the other person that picks the destination, so you can aggressively teleport people ten thousand feet high).
What is this "Travelers' League" you speak of? Is it something official or just a random name you picked?
If you had played Daggerfall, you should remember that name. It was one of the subfactions of the Mages Guild, namely the one responsible for teleporting people.
All services of the various guilds -- including the spell makers, item makers, potion makers, daedra summoners, teleporters, quest givers, healers, etc. -- had their own subfaction.
In my opinion Dagerfall's teleportation room isn't performing the teleportation, but rather makes it safe by letting you see your intended destination. Which lets you teleport (or be teleported by the mage) to a place neither of you knows without ending six feets under (or one hundred above) your destination.
No, because in Daggerfall they teleport you anywhere you want to go, including to dungeons the Traveller has presumably never visited. He just sends you where
you want to go, according to what
you want. Uses the exact same interface as fast travel, except it's instantaneous.
The teleportation chambers are just their own private offices.