Sadly, we humans can't rewrite our own code, the way a properly designed AI could.

Sure we can!
In fact, we can't stop rewriting our own code.

When you use the word "code" to describe humans, you take a certain degree of semantic liberty. So we first need to understand what is meant by "code" in this context.

In artificial computing machines, code is nothing more than a state of a chunk of memory hardware that causes the computation hardware to operate in a certain way (for a certain input). Only a tiny subset of the possible states of any chunk of memory hardware are "executable", i.e. don't cause the computation hardware to reach a certain "failure" state. This gives us an almost clear-cut distinction between (executable) code and (non-executable) data, under the assumption that data is very unlikely to be executable by chance. Given the correct design, a machine can write code to its memory and then execute it.

In humans, the distinction between memory hardware and computation hardware is unclear, if it exists at all. Moreover, it's unclear how to apply the above distinction between code and data: what is a human's "failure" state? I guess a state of the brain (containing both memory and computation hardware, until and unless we can ever separate the two) can be said to be "executable" if, placed in a certain environment, the person doesn't go and die.

It follows that any change that the brain does to its own state, which then affects its computation, to the result of not dying, is, in fact, "rewriting its own code". This, of course, happens all the time and (perhaps ironically) cannot be stopped without killing the brain.

In a wider loop, we also have drugs, medications and, eventually, gene therapy. But that's more similar to a robot reaching for the keyboard (or the screwdriver).