"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.

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. This isn't the way humans write code, and it doesn't produce an output which looks like "source code" as we know it. But it does result in programs writing programs faster, better, and cheaper than humans writing those same programs.

Regarding what AGI is "about", yes that is true in the strictest, definitional sense. But what I was trying to convey is how AGI is separate from narrow AI in that it is basically a field of meta-AI. An AGI approaches a problem by first thinking about how to solve the problem. It first thinks about thinking, before it thinks.

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.