Czuch Czuckers: It's the perfect analogy and your own logic can prove it.
A jigsaw is finished when you've discovered where each piece goes. That's just like a backgammon game is finished when the final score is known, right? Your idea. There's no point continuing to slot pieces into holes once you know where all the remaining pieces go? Why bother? You know where they go, so pack the thing away and go read a book. It doesn't have to be the last piece either; it could be the last dozen.
I hope you disagreed with that because you understand that the logical view of simply knowing where the pieces go isn't all that there is to it.
It's the same with backgammon. For some people the game ends when one player has all their pieces off the board. There's mathematically complete and there's humanly complete. Do you find it hard to understand? Yet you think that slotting known pieces pointlessly into a jigsaw makes sense?
It's the same principle but in different games. The outcome, as regards the score, maybe different from the completion. Some people see it one way. Some people see it the other.
Who are you or grenv to say that this viewpoint is ridiculous? That is simply a declaration that you have no understanding and acceptance of this difference between people. Your way is the one true way? Sure. And I wonder how many of your ways should be the one true way. [Shudder at the thought]
But the point of this discussion is not about what's the "proper" way to determine the end of a game, it's about whether you would deliberately go against the wishes of your opponent. Do you respect your opponents as people or are they just machines here to provide you with a set of postions to play?