The Pilgrimage

[Forsaken Lover]

Also, something that's always confused me--Seymour is a summoner. He is not on a pilgrimage? The game makes it pretty clear if you want to be a summoner, you go on a pilgrimage. That's the point of a summoner. Why is this?

There is some optional dialogue you can get about Seymour's backstory in Macalania Temple.
Auronlu's Complete FFX Script - Chapter VIII: Macalania

Seymour as a summoner:
As a summoner, Maester Seymour's strength is matchless. Yet, he became a maester of Yevon instead of undertaking the pilgrimage to battle Sin. I think he believes that there's more to a summoner's duty than just destroying Sin.

Consider also Lulu's and Wakka's former Summoner Father Zuke who quit his Pilgrimage and became a priest.

Seymour already has his Final Aeon in Anima. He doesn't need to go on a Pilgrimage anyway.