I imagine a group of dogs sitting around and asking "How are we so good at thinking about fun ways to play with squeeky toys?". The truth is, that our ability to reason about ourselves is limited by our ability to reason. Perhaps there are aliens out there who would laugh our cognitive abilities--their's being so much better than ours.
Less complicated systems successfully reason about more complicated systems all the time. Ditto for self-reasoning. See: bootloaders, update systems, and package managers. In order to prove that some kind of meta-cognition is inherently beyond our grasp, you don't just have to prove that the system we are attempting to reason about is more complex than ourselves, you also have to prove that the problem isn't meaningf…
What you're describing sounds like asking a PDP-11 to run GPT-3. Technically possible, in the broadest sense of the word. But a computer that can run GPT-3 successfully will look at that PDP-11 in much the same way that we look at a dog playing with a chew toy.