There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
It's a bit more complicated than that.
Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.
An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.
There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)
I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.
(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)