"Having a machine write code at the level of a human programmer" is a strawman. One can already think about machine learning techniques as the computer writing its own classification programs. These machines already "write code" (classifiers) better than any human could under the same circumstances.. it just doesn't look like code a human would write.

Yes, and my pocket calculator can compute cosines faster than Newton could. Therefore my pocket calculator is better at math than Newton.

A significant pieces of my own architecture is basically doing the same thing but with the classifiers themselves composed in a nearly turing-complete total functional language, which are then operated on by other reflective agents who are able to reason about the code due to its strong type system.

Lots of commonly used classifiers are "nearly Turing-complete".
Specifically, non-linear SVMs, feed-forward neural networks and the various kinds of decision tree methods can represent arbitrary Boolean functions, while recurrent neural networks can represent arbitrary finite state automata when implemented with finite precision arithmetic, and they are Turing-complete when implemented with arbitrary precision arithmetic.

But we don't exactly observe hordes of unemployed programmers begging in the streets after losing their jobs to some machine learning algorithm, do we?
Useful as they are, current machine learning algorithms are still very far from performing automatic programming.

But it does result in programs writing programs faster, better, and cheaper than humans writing those same programs.

Really? Can you system provide a correct implementation of the FizzBuzz program starting from a specification written in English?
Can it play competitively in a programming contest?

Or, even if your system is restricted to machine learning, can it beat random forests on a standard benchmark?

If it can do no such thing perhaps you should consider avoiding such claims, in particular when you are unwilling to show your work.

And yes, there are generally multiple ways it can actually accomplish that, e.g. the AGI could not actually solve the problem or modify itself to solve the problem, but instead output the source code for a narrow AI which efficiently does so. But if you draw the system boundary large enough, it's effectively the same thing.

Which we are currently very far from accomplishing.