Your absolutely correct observation about gaps notwithstanding, the driving question for me here is this: why does math work so well? I personally exclude any explanation that doesn't involve the Logos in some form as incoherent, but I don't have a specific answer and probably never will.
This question really really bothers me. Why can I completely describe gravitational attraction (at certain distances) by just solving for f=g(m1*m2)/r^2? It's truly disturbing when you start appreciating this for the first time.
And the ability to count sheep by counting pebbles is just a generalization of the same principles. One antenna can move in sync with another, without literally being the 'same' antenna. This is indirection or abstraction, depending on how you want to slice it.
The point being, there is no question of why arithmetical operations apply to the real world. The real world permits an infinite variety of valid and useful abstractions. I'd wager it is impossible to imagine a reality where this weren't the case.
The reason why math education takes so many years is to learn all the complexity, conventions and abuse of notation. As in speech and image interpretation, the adept cannot see the complexity.
You can make anything simple by inventing a language to state it in. Use custom entities instead of multiplying them.
Why does math work so well? Well math is pretty great but I don't think it is magical. There have been thousands of years of slow mathematical advances to get us to where we are now.
In other words, mathematics is very bad at modeling and explaining human behavior, be it in economics, history or even political science (even though one of the best political scientists that ever was, Hobbes, wrote his most famous book by trying to imitate Euclid's "Elements"). This is starting to become particularly important now because we try to build some "AI" functionalities that should imitate humans (and even surpass them) based mostly on mathematics (and some underlying data), but it is my opinion that because of this "gap" between how humans are and what mathematics can tell us about how humans are and behave, it is my opinion I say that those "AI" functionalities will never "become" human enough. Stanislaw Lem's "The Cyberiad" does a much better job compared to me at showing this gap between humans and "machines built on mathematics".
The physical observations, though, I'm still waiting on an answer from OP about that. It's fairly weasely to say something like that with no example.
And then one can ask “what is the domain of mathematics?” or even “does mathematics have a domain?”, questions which lead us into a “philosophy of science” discussion with no end in sight.
I’ve felt for quite some time that the fact that mathematics can model/answer some aspects related to physical reality is just a happy coincidence at best, which we shouldn’t insist too much upon, for fear of then risking to miss the forest because of some trees that absorb our view, like “isn’t this mathematical equation perfectly describing how galaxies interact billions of light-years away?” might obstruct from us the very dire truth that there is no math to describe what will be my cat’s movings around the room in the next 5 minutes (and it’s not for lack of trying, just look at the billions of dollars invested by hedge-funds into mathematics so that they could “model”/predict the future; I don’t think they’re scientifically anywhere close to that).
I think these are useful conversations even if we can't foresee them ending. It's what's helped us move physics beyond stuff like Newtonian mechanics where we expect things to line up with these nice equations, and apply math in a more appropriate fashion to our observations. i.e. We treat math as something we apply to our observations, and reconsider models as we run into problems, as opposed to demanding our observations line up with our initial model.
Please!