The TV coverage of gymnastics is the same video feed across the world with each TV network providing their own commentary over the top. It's always this approach for all major gymnastics competitions in qualification, team finals, and all around finals where there are multiple routines happening at once - for the first few rotations they try to show a variety, but as the competition starts to draw to a close the coverage focuses in on the top of the leader board, as most viewers want to see the routines from the top team the most.
In this final there ended up only being 0.832 between GBs 3rd place finish and France in 6th, so you can see for the people deciding what routines to show it was basically a guess as to who would end up in bronze, they guessed Italy who had been in 3rd place for much of the competition. Wrong in the end, but you can't hugely fault their logic.
Something the BBC did differently today which I think was a mistake is typically when the teams rotate between apparatus and warm-up they take the opportunity to show routines from GB that didn't make the international coverage cut. Today they favoured going to the studio to talk to Beth Tweddle, not my preference.
I think with Discovery+ and Eurosport they have a dedicated camera for each apparatus, no commentary on that, but if you really want to see GB on vault for example, you can access those feeds.