Black Box Decision problem for NFA

Suppose we are given an NFA $M$ (without $\epsilon$-transitions) that we only know the alphabet $\Sigma$ and the number of states $|Q|$ but we do not know any other details of the NFA. We want to develop a black box algorithm that can test if $L(M)=\Sigma^{*}$.
My idea is that we can feed every string up to size $|\Sigma|^{|Q|-1}$, if every strings are accepted, then $L(M)=\Sigma^{*}$. Since all strings up to size $|\Sigma|^{|Q|-1}$ must go through every reachable state in $M$, if all of them get accepted then no matter what string is input, it will be accepted. Am I correct?

Ellul, Krawetz, Shallit and Wang construct in their paper Regular Expressions: New Results and Open Problems a regular expression of length $n$ (for infinitely many $n$) such that the shortest string missing from its language has length $2^{\Omega(n)}$. Since a regular expression of length $n$ can be converted to an NFA having $O(n)$ states, this gives, for infinitely many $n$, an NFA having $n$ states such that the shortest string not accepted by the NFA has length $2^{\Omega(n)}$.

Conversely, if an NFA having $n$ states doesn't accept all strings, then it must reject some string of length shorter than $2^n$. This follows from the pumping lemma once you convert the NFA to a DFA. Hence the construction in the paper mentioned above is optimal up to the constant in the exponent.

StackExchange Computer Science Q#115371, answer score: 2