Pick up any math book and you will see each method of calculating first explained and then shown some examples and then there are assigments to work on the principle. Poomsae is not so different actually. You have the oral transmissions (wich is almost completly gone by the way). This would be the same as the explanation in the math book. For instance: Jack has two apples, Sylvia gives him two more. How many apples has Jack got now? 2 + 2 = 4. Jack has know 4 apples, but he meets his grandmother and she gives him 6 more apples. how many apples does Jack have know? 4 apples + 6 apples = 10 etc. In Taekwondo the oral transmissions works the same way. Unfortunatly much of the oral transmission is now lost or not well known to the general public. That is why you get all these weird applications as standard applications etc. You can still get the general message of old if you really study the writings of the Karate Pioneers (Choki Motobu, Gichin Funakoshi, Kenwa Mabuni, Chojun Myagi and many many more).
Poomsae is the practical examples in the "Taekwondo textbook". Here we see the motions being used combativly using sound combative principles. The pulling hand is shown in ALL the Poomsae and this means that the principle is so important and effective that we cannot fail to make use of it in combat.
Kyorigi or sparring was originally meant to be the "assigments" in the math book. Now we have the explanations (the oral transmissions of old) on what is effective in combat, we have the pracitcal examples to both understand and remember the principles in form of techniques strung together in a sequence of moves in Poomsae, and now we have sparring to test to see if everything "works". Sparring today is far removed from its intended use as it is now being developed purely for sport and physical eduacation. I am not saying this is neccesarily a bad thing as it all comes down to what you are training for, but it is however far removed from its intended usage. For instance there is no grappling or grabbing of any kind in modern Taekwondo sparring. This makes one of the most profound concepts in Taekwondo redundant (the pulling hand is illegal in sparring and so naturally it is never used outside of Poomsae).