Well lets look at chess shall we? Why does a knight move in a 2-1 "L" shape? Why not a 3-1 "L" shape? That certainly isn't "The whole game" as you put it. Now suppose here you could move your knight 3-1 as well as 2-1, would you say well it sometimes suits me to be able to do that so it's ok or would you instead pass a comment that the game wasn't being played to the rules? It is the same with backgammon. You keep asking why that is the rule. The answer to that is that if you understand the die roll possibilities and you understand that this rule applies then it is paramount as to how you move prior to the situation arising to avoid leaving yourself where having to play both dice will hurt you. Similarly you can create situations by using a backgammon tactic known as killing numbers to place your opponent in that same situation. You say it is not common, well it certainly isn't rare but what is common, though you may not realise it, is that many moves are made with the idea in mind of creating that situation, though not all will ever materialise.

To finish I would just like to deal with your comment "It would be fine if every piece could move like that through the whole game, although it wouldn't be chess." That is exactly the point, it may be fine although what we are playing certainly isn't backgammon. Having said that, without a cube in use which is far more fundamental to the game, then even if this is corrected what you are playing certainly isn't backgammon either, as cube action accounts for over 50% of the skill at the game. Regard it as a practice game for the real thing but even if you can play well here don't fool yourself you know how to play backgammon as when/if you do ever play the real game you will soon find out how little of the game you do actually understand.