Yaska's own argument against Kautsa was not the end of the story. What happened to Yaska's method afterward is its own story, and it runs for roughly two thousand years to one of the most consulted Vedic commentaries ever written.
Panini's grammatical system, composed not long after or alongside Yaska's own work, shares Nirukta's basic assumption that Sanskrit words resolve into a verbal root plus formative elements, and the two disciplines developed in dialogue with each other as parallel tools for the same underlying language, one cataloguing how words are built, the other explaining what the building blocks originally meant. Later Vedic scholarship leaned on both. The clearest case is Sayana, a fourteenth century commentator working at the Vijayanagara court in South India, whose commentaries on the Rigveda and other Vedic texts became, and largely remain, the single most widely used premodern guide to what the Vedic hymns mean, still consulted by scholars today even where modern historical linguistics reconstructs a given word's history differently than Sayana does.
Sayana draws on Yaska's etymological glosses repeatedly and directly, following Nirukta's method of deriving an obscure Vedic word from a plausible root even when the derivation is speculative by modern standards, treating Yaska's own answer to Kautsa, that Vedic language remains recoverable ordinary language rather than sound without sense, as settled ground rather than a live question. An argument mounted in Nirukta's own opening chapters to defend the very possibility of Vedic exegesis became, by the time Sayana wrote some two thousand years later, simply the unstated assumption behind doing it.