Proving theory of mind in LLMs is certainly one step toward recognizing that they are well on the path of acquiring "human-like" intelligence - if there are still people around who doubt that (proof of which would be if the ensuing discussion centers around if this is still just guessing the next token or whether hallucinations are proof of permanent unintelligence, etc).
Theory of mind is obviously coupled with the recognition that we can be outsmarted - so this type of research is crucial in order to gauge what stage we are on.