While pressing on the clutch pedal, the clutch plate is staying engaged. I have watched the slave cylinder and it is moving.
Even if the clutch slave cylinder moves, it doesn't mean it's moving enough. If the hydraulics need to be bled, the clutch won't disengage and the friction disk will be dragging on the pressure plate and flywheel. What you are describing is indicative of the clutch needing bled.
EDIT: Considering what you said in your comment, it means you need either a new master or slave (or both). One of the two is leaking back after the pedal is depressed. One of them is worn out. When pressure is applied, the fluid leaks past it and the slave goes back into position. If I were you, I'd replace them both and call it a day. Remember, you'll have to bleed them once they are changed out. I bet you are a pro at that by now.