Throughout most of the math education debate is the idea that hard work by teachers and math education researchers is good, but work by math foundations people is bad. Math foundations gets no respect in math ed. Why is that?
Part of it is fear. Math ed people for the most part and most teachers do not know math foundations. They were not taught them in college.
In the US, the Peano Axioms are not typically taught to pre-service teachers. As we know, at least once they were taught to them at PH-Heidelberg.
Part of it is the failure of 1960s era New Math reforms. That gave math foundations a bad name. The main problem with 1960s New Math was it did not push the Peano Axioms and the definition of addition and multiplication of natural numbers.
New Math pushed set theory and maybe a few other things, but it failed to push the road from counting in the Peano Axioms using the successor function through defining addition and multiplication of natural numbers and then proving the properties of these functions.
Part of the reason for this was the Edmund Landau Foundations of Analysis book used the Kalmar definition of addition, and this definition is hard to understand. That was the main book in the 1960s on Peano Axioms. That book also covers up the important sequence in the Dedekind book through order of natural numbers, the recursion theorem and then the definitions of addition and multiplication recursively.
The meaning of elementary math is in math foundations and there alone. The meaning is not applications. Telling students implicitly that the meaning of natural numbers or fractions is the applications is misleading because it is wrong.
The short road through elementary math is the road of abstraction. That road is laid out in the Dedekind 1888 book. The Dedekind book lightens the cognitive load because it contains the path through the very beginning of elementary math. However, it takes hard work itself to learn. It lacks examples and exercises so it is not suitable for self study by most people.