It was within the rules, so Bairstow was out.
When a batter plays and misses a shot it's easy to get off balance and self preservation has you putting your feet where you don't really want them to stop you ending up on your arse. It's not unusual for a keeper to run the ball at the stumps in that situation and if you are successful it's a fair stumping. A keeper standing up will also often wait for the batter to come fully to rest. If the batter inadvertently raises a foot in that case and is stumped that is also a fair stumping. It happens all the time. I've seen lots of video posted of Bairstow doing exactly that over the last few days, but this is a completely different thing and is just a fair stumping. The people posting them don't understand the game.
The reason people think Bairstow's dismissal is different is that he is not off balance or inadvertently raising a foot. He taps his foot back into the crease, which is the same as tapping your bat into the crease. It shows that you are under control and not attempting a run. The umpire at the bowlers end (the one he is looking at) is reaching down for the bowlers cap and starting to turn towards square leg. 99.9999% of the time that would be enough for batsmen and fielders to think the ball is dead and the over complete. But, the ball was already on it's way towards the stumps at that point which means it wasn't dead - it was still in play. It hit the stumps and Bairstow had walked so he was out.
The Aussies have changed the game over the years with co-ordinated appealing, not walking and many other things. It took other nations years to catch up and when they did - guess what? - the Aussies were the first to complain. It was a National crisis in Aussie when Stuart Broad didn't walk once, which was completely unfair because the Aussies had already 'not walked' on about 3 or 4 occasions in the same match and also did so another couple of times afterwards. They still bitched about it for years after though, so they ain't as sweet and innocent as they are making out.
When a batter plays and misses a shot it's easy to get off balance and self preservation has you putting your feet where you don't really want them to stop you ending up on your arse. It's not unusual for a keeper to run the ball at the stumps in that situation and if you are successful it's a fair stumping. A keeper standing up will also often wait for the batter to come fully to rest. If the batter inadvertently raises a foot in that case and is stumped that is also a fair stumping. It happens all the time. I've seen lots of video posted of Bairstow doing exactly that over the last few days, but this is a completely different thing and is just a fair stumping. The people posting them don't understand the game.
The reason people think Bairstow's dismissal is different is that he is not off balance or inadvertently raising a foot. He taps his foot back into the crease, which is the same as tapping your bat into the crease. It shows that you are under control and not attempting a run. The umpire at the bowlers end (the one he is looking at) is reaching down for the bowlers cap and starting to turn towards square leg. 99.9999% of the time that would be enough for batsmen and fielders to think the ball is dead and the over complete. But, the ball was already on it's way towards the stumps at that point which means it wasn't dead - it was still in play. It hit the stumps and Bairstow had walked so he was out.
The Aussies have changed the game over the years with co-ordinated appealing, not walking and many other things. It took other nations years to catch up and when they did - guess what? - the Aussies were the first to complain. It was a National crisis in Aussie when Stuart Broad didn't walk once, which was completely unfair because the Aussies had already 'not walked' on about 3 or 4 occasions in the same match and also did so another couple of times afterwards. They still bitched about it for years after though, so they ain't as sweet and innocent as they are making out.