Does that rhyme in Turkish? I’m curious now if it has a certain “beat” when it’s spoken that sort of helps it persist as an idea in the language (at least that seems to). Sort of a short form poetry.

Yes it rhymes. Turkish words are always read as they are written so if it's the same letters it'll sound the same.

The way we were thought about these in school regarding proverbs and idioms is that they were distilled through time to be their most economical and impactful form. So yes, the good ones rhyme and are either funny or poignant.