Hm who cares about utility in this case?

They're saying you can't find a ruler accurate enough to be sure the number you measure is sqrt(2) and not sqrt(2) for the first 1000 digits then something else. And eventually, as you build better and better rulers, it will turn out that physical reality doesn't encode enough information to be sure. Anything you can measure is rational.

A perfect mathematical square cannot exist in reality.

You don’t need quantum mechanics. Trigonometric functions are everywhere in classical mechanics. Gaussians, exponential, and logs are everywhere in statistical physics. You cannot do much if you don’t use transcendental numbers. Hell, you just need a circle to come across pi. It’s rational numbers that are special.

They're accurately modeled. Just as Newtownian phenomena are accurately modeled, until they aren't. Reality is not necessarily reflective of any model.

Consider the ideal gas law: pV=nRT

Five continuous quantities related to each other, where by default when not specified we can safely assume real values, right? So we must have real values in reality, right?

But we know that gas is not continuous. The "real" ideal gas law that relates those quantities really needs you to input every gas molecule, every velocity of every gas molecule, every detail of each gas molecule, and if you really want to get precise, everything down to every neutrino passing through the volume. Such a real formula would need to include terms for things like the self-gravitation of the gas affecting all those parameters. We use a simple real-valued formula because it is good enough to capture what we're interested in. None of the five quantities in that formula "actually" exist, in the sense of being a single number that fully captures the exact details of what is going on. It's a model, not reality.

Similarly, all those things using trig and such are models, not reality.

But while true, those in some sense miss something even more important, which I alluded to strongly but will spell out clearly here: What would it mean to have a provably irrational value in hand? In the real universe? Not metaphorically, but some sort of real value fully in your hand, such that you fully and completely know it is an irrational value? Some measure of some quantity that you have to that detail? It means that if you tell me the value is X, but I challenge you that where you say the Graham's Number-th digit of your number is a 7, I say it is actually a 4, you can prove me wrong. Not by math; by measurement, by observation of the value that you have "in hand".

You can never gather that much information about any quantity in the real universe. You will always have finite information about it. Any such quantity will be indistinguishable from a rational number by any real test you could possibly run. You can never tell me with confidence that you have an irrational number in hand.

Another way of looking at it: Consider the Taylor expansion of the sine function. To be the transcendental function it is in math, it must use all the terms of the series. Any finite number of terms is still a polynomial, no matter how large. Now, again, I tell you that by the Graham's Number term, the universe is no longer using those terms. How do you prove me wrong by measurement?

All you can give me is that some value in hand sure does seem to bear a strong resemblance to this particular irrational value, pi or e perhaps, but that's all. You can't go out the infinite number of digits necessary to prove that you have exactly pi or e.

Many candidates for the Theory of Everything don't even have the infinite granularity in the universe in them necessary to have that detailed an object in reality, containing some sort of "smallest thing" in them and minimum granularity. Even the ones that do still have the Planck size limit that they don't claim to be able to meaningfully see beyond with real measurements.