In proving Gödel's incompleteness theorem, why does he needed the Diagonal Lemma or the Fixed Point Theorem for building a formula $\phi$ that spoke about itself? Can't this formula be built this way:
Diagonal Function. Let $D(x, y)$ be the diagonal function, such that $D$ returns the result obtained by replacing the formula with Gödel number $y$ for all free occurrences of $a$ in the formula with Gödel number $x$.
Example. Let $\psi(a)$ be a formula that affirms that some formula with the Gödel number of $a$ is closed (has no free variables) and $k$ be it's Gödel number. Using Diagonal Function to construct a formula, by applying $D(k, k) = j$, the Gödel number $j$ will already be the Gödel number of a formula affirming that the formula itself has no free variables. I mean, $\ulcorner \psi(\ulcorner \psi(a) \urcorner) \urcorner = j$, or $\ulcorner \psi(\overline{k}) \urcorner = j$.
What i am missing here?
If I understand your notation correctly, the formula (whose Gödel number is) $j$ does not assert that $j$ itself has no free variables, merely that $\psi(a)$ (which is different from $j$ itself) has no free variables. Which, incidentally, is false because it has $a$ as a free variable.