There are well-formed statements that can be proved but which assert that its godelized value represents a non-provable theorem.
Therefore, you must accept that it and its contradiction are both provable (leading to an inconsistent system), or not accept it and now there are provable theorems that cannot be expressed in the system.
Furthermore, that this can be constructed from anything with base arithmetic and induction over first-order logic (Gödel's original paper included how broadly it could be applied to basically every logical system).
The important thing to note is that it doesn't have anything to do with truth or truth-values of propositions. It breaks the fundamental operation of the provability of a statement.
And, since many proofs are done by assuming a statement's inverse and trying to prove a contradiction, having a known contradiction in the set of provable statements can effectively allow any statement to be proven. Keeping the contradiction is not actually an option.
There are well-formed statements that can be proved but which assert that its godelized value represents a non-provable theorem.
Therefore, you must accept that it and its contradiction are both provable (leading to an inconsistent system), or not accept it and now there are provable theorems that cannot be expressed in the system.
Furthermore, that this can be constructed from anything with base arithmetic and induction over first-order logic (Gödel's original paper included how broadly it could be applied to basically every logical system).