AIUI, the more meaningful interpretation of the saying is about exceptional achievement, not basic competence.
I.e., “Those who can perform well enough to make the next cut of elite performers will become elite performers; those who can’t will teach their less outstanding skills to the next generation of aspiring novices.”
The point is that every art or discipline needs more instructors than virtuosi or megastars, and so the competition and rewards for being an instructor are typically less than those for being a megastar. And consequently, the most gifted and accomplished practitioners become the megastars, and the lower tiers become the teachers.
But that’s not at all the same thing as claiming that somebody who undertakes to teach a subject is by definition absolutely incompetent at it themselves. Which would be fundamentally nonsensical, in addition to being directly contradicted by almost everybody’s personal experience with some highly accomplished teacher(s) of some subject(s).
That said, I agree on the recommendations to the OP about getting voice coaching to help smooth out his passaggio, and I think he’ll probably get best results by shaking the grapevine. OP, as @fachverwirrt suggests, ask the singers you know and admire about how they were trained and whom they might recommend, or ask a trusted voice teacher/coach elsewhere if they could connect you with a colleague in your local area.