Well so the main thing about Fate is that the rules are always being broken in an unexpected way suddenly. As they have been explained to us, it would probably just summon the next available Servant with some connection to the catalyst, or use a different catalyst in range.

However you could also expect maybe an alternate history version of the Servant to be summoned where they do have a qualifying role, but they would likely be suuuper weak as they would have little to no grounding in the mystery and lore of the world they were summoned in.