You must select stars 3-6 to spawn the owl.
Momentum is stored and reversed when you jump onto the owl so try to land at the spot in the video and jump directly up at him with no directional input. Turn in the tree before jumping out by holding right for a little bit.
Select the 1st star to not spawn the owl and avoid lag from the tower.
Hold slightly up after landing on the spinning platform to slowly turn upwards towards the cage.