You can always arrive at a correct statement. A random expression generator can occasionally do that. You just can't tell if it is true.
A simple integer counter can generate the Gödel statement, it just doesn't have the ability to identify it.
You could take a guess, which is what people are doing when they say they they understand, they have applied hurisics to convince themselves, the problem is either decidable, or they are simply wrong.
But in the Penrose argument, we can start from a true system and use reflection to arrive at another true statement which is not deducible from the original system.
This is important to the argument as one starts with a proposed program which can perform mathematical reasoning correctly and is not just a random generator. Then, the inability to see the new statement is a genuine limitation.
How is it something that a computer cannot do? It seems to be just convincing yourself.
Gödel's theorum itself does not help you here because you are trying to identify a undecidable but true. Gödel only showed that there are undecidable true statements, not what they are.
Godel explicitly constructs the statements (which are essentially encoding a particular self-referential statement into logic). Penrose makes this more concrete where given an accurate, but necessarily partial procedure for detecting when a program halts, one can use diagonalization to show that a program wont halt even though the earlier process couldn't detect it.
This construction is something that a computer can do, but not the original system itself. Once you augment the system, there is now a new statement it cant see and so on. (so on, here involves ordinals).
There is no special insight required here. Merely, a simple diagonalization argument (https://news.ycombinator.com/item?id=43257904) which given a partial process to detect if a Turing machine halts, constructs a new machine which does not halt but was not detected by the earlier process.
