I’m going with “not a new problem”.
Every era has had to deal with the issue of unfakeable authentication. Royal seals, signet rings, expensive stationery (like vellum) all made the process of authenticating the genuine will of the King sufficiently secure for functional purposes, even though the possibility of forgery existed. Part of the proof of authenticity was not just the bells and whistles attached to the document itself in isolation, but the context of the document. For any Royal proclamation, there are likely to have been discussions and prior correspondence in the history of which the decree in question is embedded.
And there was always the medieval equivalent of blockchain: if you receive an apparently odd decree, you can send a trusted person to ask the King if this is real or not.
Transpose those ideas to the modern era. In the arms race of authentication technology, we have for a short period lived in an era where video did not seem to be convincingly fakeable. If they become so, then we will have to resort to older skills in assessing authenticity: source, inherent probability, checking surrounding info to authenticate, etc.
Thus, if we have a video of The Donald and Chuck Schumer announcing to the media that they were running away together, then we can check diaries to see what they were doing when the video was supposedly made; we can check with those with who were supposedly there, etc.
For a long time now it has been possible to make convincing fake still photos, but the sky hasn’t fallen. There was a transition period prior to which fake photos had been unthinkable. Naivety about the supposed unfakeability of photos led to the embarrassment of Arthur Conan Doyle over the Cottesley Fairies. But if photos were to emerge of The Donald blowing Prince Harry to get a wedding invite, no scandal would arise. No-one would believe it. It would fail context tests - when did this happen? What do we know those two to have been doing at the time? Who was the photographer? Why was he allowed to be there? And so on.
Re the prospect of drawing the teeth of scandalous real videos by flooding the world with fakes - did that happen with photos? I don’t recall such a thing.
As I say, we have been briefly spoiled by the illusion of unfakeable video. But the arms race goes on. World keeps turning.