The TV coverage of gymnastics is the same across the world with different commentary over the top. It's typically this approach for all 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 people 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.
On the BBC Olympic coverage in general sadly their hands are very much tied with the horrible situation of only being permitted two live streams, they are always going to upset someone and they need to be dynamic, to show a variety of sports and British medals. I'm angry as we didn't just lose the bid, it was a closed door deal and BBC were not even given the opportunity to keep the coverage. For me making it so inaccessible goes against the spirit of the Olympics, and is very sad for our athletes who aren't getting the coverage they should.