Torah She’bal Peh was not written from the time of Har Sinai through the Tannaic period. How many centuries is that? And even then it was written sparsely, not all of the Torah She’bal Peh was written. In the Amoroic time more of the unwritten was written. And still not all of it was written. Much was still passed down verbally.
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And more was written down during the period of the Rishonim, but still not all. etc. But all this is only tangential to the point. Mesorah has always been a feature of Judaism. A feature, not a bug. And a verbal (i.e. unwritten) mesorah is something that always existed. Especially insofar as they concern minhagim. Minhagim have always and frequently been unwritten. Especially until relatively recent centuries. So a lack of a written text proves not a whole lot other than it wasn’t written. (Or that it may have been written but the seforim containing it has been lost over the centuries as so many seforim have been lost to the ages.)