I've just watched it again... WB is giving constant commentary of what he is seeing on the replay being shown to him, and his words all perfectly match the replay shown on TV, so it seems unlikely that he was watching something different while accurately describing what we were seeing on the TV.
The contentious clip was shown on TV a few seconds before the restart kick. I have no idea if it was ever shown on the big screens.
The broadcaster sends their pictures to live TV.
They also make them "available" to the host club to display on the big screen.
Most of the time the host club will show the live tv picture, but where unasked for replays are being shown of an incident some clubs have been known to put an advert up rather than the live feed.
It would be very foolish however for a club to display anything except the live feed when a TMO review is underway, as they would be in trouble with the competition organiser, but it has been known to happen.
The TMO will see the live feed (as well as all the other camera feeds), but wont know what the ref is seeing on the big screens as that is controlled by the club. He will assume that the ref is seeing the live feed, but wont know for sure.
I'm not saying that is necessarily what happened here, just correcting your blanket statement that what the TMO sees is what the ref sees.