Fede wrote:tl;dr: at peak time it's already hard being on top of everything as it is now. I really couldn't add checking 20 webcams

And besides, I suspect my Internet connection wouldn't bear the load, unless they streamed at 144p.
A few random thoughts:
1. Webcam feeds don't necessarily need to be checked in real time. Seems like they could be saved and reviewed later. However, watching them later raises the possibility that one discovers too late that there was a technical problem -- like the camera being improperly positioned. It might be necessary to at least confirm that the feed is being correctly recorded at the start and then check in periodically to make sure that nothing has gone wrong.
2. The other participants in the tournament have a vested interest in this monitoring, and that could be utilized -- particularly, opponents. You could, for example, make each player responsible for ensuring that their opponent's feed is properly set up and recorded. Might even make sense to make them responsible for reviewing the feed after the game. They have a vested interest in the outcome of the monitoring, but a panel of neutral arbitrators could screened out false accusations.
3. I suspect that if this strategy of observing players gets used, there would need to be a very detailed and carefully laid out description of what players are and are not allowed to do during the game. Headphones seems clear, but there are probably other issues that need to be addressed. For example, what if someone needs to leave the room to get a drink or to use the bathroom? How should the camera be positioned -- to just observe the player or should the entire room be visible? If the latter, does the player need to be alone? Someone mentioned on another thread that the Chess world does this regularly, it might be good to review their procedures as a starting point.
That said, this strategy can only go so far. Players, for example, could have an accomplice in another room follow the game and relay messages back either orally or in writing from outside of camera shot.