However, the key isn't necessarily video recordings. It's deliberately practicing individual things you are bad at in isolation until you are good at them.
We need to record LoL and StarCraft games because it's tough to pause and redo something in the middle of a match. Similarly recording baseball swings or people dancing or swimming is useful for reviewing the complete action.
But when you practice piano, you can pause and redo a measure or line until you get it right, then slowly reincorporate it as part of the whole piece.
The WORST thing you can do is keep playing and think "I'll fix that later." Because inevitably you'll play a piece again, probably do the same mistake, and now you're actually practicing the mistake and ingraining it into your memory.
A video recording and reviewing it of a whole piece can help you go back and isolate your mistakes though. But it's just a tool for the goal of identifying and fixing mistakes.
We used an iPad to record swings of our little leaguers and it made a huge difference for them.
It helps because it’s easier to see what is happening vs hearing it. One example that we would see is kids dropping the back shoulder when they swing. Once they see it, they correct.
My wife had trouble with a certain move in dog agility until she saw herself do it on video and it was clear what was wrong.