What is wrong with the following argument?
Let $F = T + ¬G_T$, where $T$ is an effectively generated formal system and $G_T$ is its Godel sentence.
Then, it is possible to prove within $F$ the First Incompleteness Theorem for $T$: $Con(T) \Rightarrow G_T$, just by utilising the axioms for $T$.
But from this the contrapositive follows: $¬G_T \Rightarrow ¬Con(T)$.
So in F it is possible to prove $¬Con(T)$, since $¬G_T$ is an axiom.
But it then follows that $¬Con(F)$, since the same contradiction that is derivable in $T$ is also derivable in $F$ by restricting ourselves to the axioms for $T$.
So any formal system that is of the form $F = T + ¬G_T$ is inconsistent.
Not exactly. You have shown that $F$ can formally prove $\neg \text{Con}(F)$ but you can't deduce from it that $F$ is inconsistent (which is a meta-statement). You seem to believe that the following meta-theorem is true :
"If $F \vdash \neg \text{Con}(F)$ then $F$ is inconsistent"
but how would you prove such a thing ?