Those transcriptions are already done by LLMs in the first place - in fact, audio transcription was one of the very first large scale commercial uses of the technology in its current iteration.

This is just like playing a game of markov telephone where the step in OP's solution is likely higher compute cost than the step YT uses, because YT is interested in minimizing costs.

Probably just "regular" LMs, not large LMs, I assume. I assume some LM with 10-100M params or so, which is cheap to use (and very standard for ASR).