Peano was very much into symbols and linguistic games. His book in 1889 was really a big step backward in confusing the issues from Dedekind in 1888. Dedekind had a major flaw in the existence of an infinite set.
However, once Peano confused the issues, in particular the logical order of Dedekind of closed sets under some transformation, intersection arguments, tail sets, order, recursion theorem, and then addition, it never got put back together.
David Joyce and David Groisser are two standouts. But the profession is so badly flummoxed it can’t even recognize their contribution. It just ignores them. This is the professional research mathematicians not just teachers or math ed professors.
If our research mathematicians can’t recognize the problems in the more typical approaches to the Peano Axioms and recognize the superior approach of Dedekind, is it any wonder that the entire elementary school curriculum is just a random succession of drills that have no explanation as to why it is done that way?
New math failed in the 1950s and 1960s because the people doing it did not know how to tie set theory to arithmetic from counting through fractions. It wasn’t the children or teachers who failed, it was the research mathematicians who had become disoriented by post Dedekind texts into not understanding order of natural numbers.
It is like they worked for a government agency. Which many of them did. The 19th century reached a high point in understanding numbers in 1888, and from there mankind actually went backwards even among our top minds in math.
This is the root of the random drill nature of elementary math. The logical progression in Dedekind 1888 was forgotten. New Math in the 1950s and 1960s failed because it did not understand it.
Until this is fixed, school math will still seem like a daily drill assembly line that has no reason as to what is done or why it is in the order it is.