I remember having played something like this before. It was called Expand. It has to be played with an analog stick. There are keyboard controls, but these are useless due to the circumferential puzzles.
(1:50) That 1-tap solution can be achieved by having the rotation begin in the direction such that when the dot gets launched to the outer platforms, gravity does the work instead.
https://youtu.be/kKomJqtm7J4