While this is true, it is surprising that is it impossible to engineer a system to be complete and consistent about arithmetic, as opposed to systems which are not engineered well, like your 2nd paragraph.
If I take "1 is even", "1 is odd" and "no number can be even and odd" as my axioms, then there is obviously a problem, but of my own doing.
Russel and Whitehead's system is well-engineered; that's why it doesn't prove a contradiction (presumably) while your badly engineered system does. The issue is that R&W made their system powerful enough that it's also incomplete, due to the fact that their system contains smartass statements like G that are hell-bent on creating paradoxes if given the chance.
Before Godel's time, people just didn't have enough experience with computers to realize that you have to deal with code injection attacks every time you try to build a powerful platform of any kind. In this case, Godel Numbering is the hack that allows code injection into a formal system that's supposed to just be highly insightful about properties of the infinite world of natural numbers.
Russell's Principia Mathematica (PM) is indeed better
engineered than its predecessor by Frege because it has orders
on propositions. Because of orders on proportions, PM does
not allow the [Gödel 1931] proposition I'mUnprovable.
Furthermore, adding the proposition I'mUnprovable would
make PM inconsistent.
The Gödel number of a proposition in PM is itself
"incomplete" because it *doesn't* include the order of the
proposition. Allowing its Gödel number to represent a
proposition is indeed a kind of "code injection" attack,
which if allowed would make PM inconsistent.
If I take "1 is even", "1 is odd" and "no number can be even and odd" as my axioms, then there is obviously a problem, but of my own doing.