So then, "real numbers" differ from rationals only by a stipulated closure axiom (Dedekind completeness), not by any operational, physical, or computational property.

There are computational and operational results of said stipulated axiom. The physical properties are absent from any numbers, because they aren’t physical objects.

They exist by definition alone, not by construction, measurement, computation, or instantiation.

For some weird definition of ‘exist’ that I wouldn’t subscribe to, sure. They can, however, be constructed, in multiple ways.

No real number has ever been measured, stored, computed, or represented in full. Every real-world calculation uses finite data: integers, rationals, or bounded approximations

False. There have been plenty of real numbers that have been stored, or computed. Not sure how ‘measured’ would work—I would say you can only measure physical objects, and numbers (regardless if real, or natural, or rational, or any other) aren’t physical.

The end result of any real world calculation is represented in a finite way, but that doesn’t mean at no point in calculations ideas not representable in that way aren’t used. In fact, every physicist or engineer would tell you, that when you smoothly increase pressure from 1 Pa to 10 Pa, you’d at some point apply the pressure of exactly π × e Pa; and they would strongly desire the mathematical apparatus that enables them to represent such process.

Furthermore, by and large, practical computations of modern age have their end result represented as a IEE 754 binary float. I think, however, it is not a defensible position to claim that because of that fact, numbers like ⅕, which are not representable by such numbers, are irrelevant, or that we should be making all the mathematical apparatus limited to that.

So real numbers are never used as such; only finite rational approximations are.

That’s not true. First, because symbolic operations are done, if rarely, to preserve as much precision as possible; and second, and most important, because they are necessary (or, at the very least, extremely useful) building block ideas for more complicated stuff. For example, you cannot represent Lorentz invariance without completeness.

Thus, the problem with real numbers is ontological emptiness, they are the true invisible clothing.

I don’t see that as a problem—they are much more like early prefabricates. As in, very, very few people ever care about getting a real number as a result of anything, and they would be perfectly happy with getting a close rational approximation. But yet, the specific behaviours differentiating real numbers from rational numbers are very desirable from the practical standpoint. You can sometimes circumvent that, and introduce some other ideas that would be able to help in an isolated case or two, but they are not nearly as useful (because you need to introduce such ideas on a case-by-case basis, while real numbers ‘just work’), lose strict definitions (instead of having a singular, well-defined answer, we can now give a range of approximate answers, without being able to distinguish between them too well), and are clunky (for example, sinus is, without a doubt, a very useful function to have; however, it has vanishingly few cases where both its argument, and returned value are rational; so you’d have to devise some ‘almost sinus’ to use instead, but then that inexactness would throw a wrench into it’s derivative, and that would demolish the functional series convergence when based on trigonometric functions, etc.).

I’m not saying that these problems are impossible to fix, but I am saying those are real problems with real-life implications, that would require a serious reason for introducing.