Polynomially many times launch $NL$ machine - is it in $NL$? On example of ACYCLIC
You cannot implement this part in NL:

nondeterministically check something
if yes, return no

Consider the following analogy: the language $K$ of all Turing machines that halt is recursively enumerable but its negation isn't. That is, $K$ is accepted by some nondeterministic Turing machine, but its negation cannot be accepted by any nondeterministic Turing machine. This means that you provably cannot implement the following using nondeterminism:

nondeterministically check if the given Turing machine halts
if yes, return no

Your idea suffers from the same problem.

Here is a different angle. The following algorithm checks nondeterministically that $s$ and $t$ are connected:

guess a path $w$ from $s$ to $t$
verify that $w$ is a path; if so, output yes, otherwise, output no

The semantics of this algorithm are as follows:

If there is some witness that causes the algorithm to output yes, then the input is a yes input.

If all witnesses cause the algorithm to output no, then the input is a no input.


What you are trying to do is as follows:

guess a path $w$ from $s$ to $t$
verify that $w$ is a path; if so, output no, otherwise, output yes

All inputs are yes instances in this case, since there is always some witness $w$ that isn't a path from $s$ to $t$.
In order to check that there is no path, you would have to do the following:

go over all possible paths $w$ from $s$ to $t$
if none of them works, output yes, otherwise, output no

Unfortunately, this doesn't describe a nondeterministic Turing machine.