Codec vulnerabilities in the browser have been a recurring source of exploits, as have similar vulnerabilities in phone messenger apps. The phone ones are particularly bad because phones typically preprocess received media files (for thumbnailing etc) so a vulnerability here can sometimes be chained into a remotely triggered near-instant compromise.

I don’t know if there is data on exploits due to downloaded media but it would be an easy way to exploit specific target populations (find a video of interest to them and “leak” it somewhere).

If said videos or media are doing that, there would be a corpus of samples by now. I'm very sceptical of this.

Absence of evidence is not evidence of absence, but I concede that attacks via locally downloaded video may not be a widespread problem at the moment. The problem is we just don’t know. I doubt these attacks would be effective against government or corporate targets (not sure who is torrenting on their work computer) and that’s where most post-exploit forensic data is gathered.