$L_{\Sigma^*}=\{\langle M\rangle|L(M)=\Sigma^*\}\notin coRE$
You say

$\langle M \rangle$ should accept all the inputs, and if one of the inputs rejected it's not $L(M)=\Sigma^∗$.

The above two statements are true, but you are not handling all behaviors of $M$.
In particular you are not considering that $M$ could never halt on some input $x$. In that case $L(M) \neq \Sigma^*$.
In short, your observation do not imply the existence of a Turing machine that accepts on input $M$ whenever $L(M) \neq \Sigma^*$, which would show $\overline{L_{\Sigma^*}} \in \mathsf{RE}$.
In fact, you can formally prove that $\overline{L_{\Sigma^*}} \not\in \mathsf{RE}$ by reducing from the halting problem:
for any Turing machine $T$ you can construct another machine $M_T$ that simulates $T$ and then accepts.
If $\overline{L_{\Sigma^*}}$ were in RE, then there would be a Turing machine $T^*$ that accepts iff $M_T \in \overline{L_{\Sigma^*}}$, i.e., iff $T$ does not halt.
Interleaving the execution of $T^*$ with that of $T$ itself allows  you to solve the halting problem.