I think this is one of the places where we get a messy divide between the single-player story and collective MMORPG experience.

We-the-singular-Warrior-of-Light (and increasingly friend to all four high houses, depending on which side quests you’ve run) could probably be accepted into the Temple Knights if we wanted to be.

The hundreds of other adventurers who are not the Warrior of Light in your version of the story? Not so fast. They haven’t proven themself like you did.

It’s not impossible for them to set up a ‘GC wing’ of the knights in future, with looser standards on who can join. Although it would potentially get messy lore-wise to have it there in plain sight for new players coming up through Heavensward.