Yes, cats care about closed doors as they get in the way of their day-to-day acitivities without any perceived rhyme or reason. The problem is that cats don’t really understand doors. They are a human invention designed to compartmentalise places. Instinctively domestic cats don’t see the world as compartmentalised. When they are closed, doors are meant to create privacy. When locked they provide security.
Cats are familiar with the concepts of privacy and security as they establish a den when birthing kittens. But the total barrier that closed doors create are alien to them. I would say that cats do ‘care’ about doors in the sense that they are a nuisance. They’d be happier without them. Because they are a barrier to normal activities such as going outside they have learned to ask their human companion to open them on demand if there is no cat door. Their request is a learned process too. The carefully produced meow is designed to tweak the human sentiments. Some cats have learned to produce a baby-like meow for added impact on their human’s emotions.
Also, as you are probably aware cats sometimes learn to open doors. They learn by observation. And I am sure your cat often pushes a partially closed door open. Cats have certainly learned to live with this strange human invention.
Cats are adaptable like that; they find a way to get through those damnable doors. There are numerous other aspects of the human lifestyle that domestic cats have learned to live with and adapt to. There is a good argument for inside doors to have cat doors as shown in the picture on this page.