I'm not confused about the difference between what an irrational number is and the cardinality of the real numbers and it's various subsets.

It all depends on one's philosophy and existence. When you say, "only countably many of them exists....The rest are just..." you are admitting that at least in some sense they all exist. Indeed, one must be careful when making a statement, "irrational real numbers exist in nature". You can't point to anything and say, "There is the number square root of 2." It does not exist as a physical object.

Even if one were to make a giant philosophical leap and believe that one can make right triangle whose legs have precisely the same length (down to the last particle!) you don't have the existence of an irrational number. You have the hypotenuse of a triangle. The number itself does not exist in the physical sense.

Numbers are concepts and to say that one such concept exists but not another because there is some physical quantity is "equal" to it in former case but not the latter is an absurdity in my opinion. There are errors in all physical measurements. How is one going to distinguish between which numbers exist in your sense and which ones don't when the numbers are close enough together that no measurement can possibly distinguish between them?

The perfect diagonal of a square is an ideal. Actually measurable diagonals of squares may only be able to have a finite set of lengths. Note that this may imply (perhaps: does imply) that ideally square angles do not exist. I'm not bothered by those things :)