Well that would be true, except Kirei and Gilgamesh have one of the most unique Master/Servant relationship in Fate. After Fate/zero Kirei should’ve died however due to Gilgamesh being replenished thru the Grail’s contents, part of that rejuvenation is given to Kirei thru the Servant bond. At this point Kirei’s life is now dependent on Gilgamesh, rather than the other way around.

Heaven’s feel spoiler This is further shown thru the fact the Kirei doesn’t possess a “heart” and begins to deteriorate after the “Makiri grail” finishes Gil for good. Hence why the grail never bought Gil out as a shadow yet bought out most of the others.

So yeah Kirei’s existence is weird, because of his circumstances, I’m assuming the Grail would no longer acknowledge him as a master, meaning he needed someone else to be chosen and summon a servant for him. Maybe I’m wrong but that’s what I understood from the VN and anime.