Tysm for your feedback! The idea is that you have paths open, but need some item to succeed. For the Gears minigame: They are noisy. To make them more quiet you need to do the Cooking mini-game first. And to go through the marching knights you need a helmet to look like them.
So the order to complete the game would be: Cooking -> Gears -> Kitchen Basement -> Descent into the dried well (Main Hall) -> Marching Knights -> Catch the Fairy.