Why proving that two languages used to merge into a regular language are not necessarily regular isn't possible with closure properties?
Your construction outputs the language
$$ \{ w \in L_1 : \exists z \in L_2 \text{ s.t. } |z|=|w| \}. $$
In words, it outputs those words in $L_1$ whose length matches the length of some word in $L_2$. In general, this could be much smaller than $L_1$.

In particular, if $L_2$ is a finite language, then the result of merging $L_1$ and $L_2$ is finite, and so regular, for every language $L_1$.