Lots of good input. But the anticipation thing should not be underestimated. Over and over again, in that video, you are responding late to what is coming at you. You can't use good footwork if you start out late to the ball. If you start moving to the ball after it is on its way to the net or over the net, there is no way you can use good footwork when you are nowhere near the ball. A lot of the issue is about brain-processing. Even with the shots that are directly at you or near enough to you so footwork is not an issue, you are responding to the ball only after it is nearing the net or over the net.
Watch your opponent's serve at 1:12. You see where the ball is going pretty much as he is serving and then you stand and wait, and then you move when the ball is on your side, and you footwork for that ball is what it should be but you get there late, and of course, your racket gets there late. It may not be all of why you miss the ball. But your racket was not there in time so, even if you had read the spin well and were adjusting to what was on the ball, your racket was there so late that it is hard to imagine that you would have made a quality return if you had read the spin well. In other words, even if you returned that serve, you should be looking at how slow you were to get in position to receive it.
So, a lot of the issues seem to me to be about anticipation, reading the play, and reaction time. And, to me, it does look like you are seeing what is happening and then there is a gap between seeing and you responding.
But Tony's recommendation and the type of multiball drills you do should get your brain processing working faster and get you seeing the play faster. So, it would fix the technical details and it would help improve the brain-processing issues that are causing you to be responding late to what you are seeing from your opponent.