Is it always enough to prove that a subset $L'$ of a language $L$ is unprovable in order to prove that $L$ is undecidable?
No, your conclusion is not correct. The fact that $L'\subseteq L$ is undecidable do not imply anything about $L$. Indeed, consider $L=\Sigma^*$
for some finite alphabet $\Sigma$. Then, $L$ is decidable.

Now consider $L'\subset L$ such that $w\in L'$ iff $w$ is the description of a halting Turing machine written with characters from $\Sigma$. Then, $L'$ is, of course, undecidable.