Thank you very much for the detailed answers. My idea for a gimbal was to get clean camera movements, i.e. that the image is in balance and that scenes can be captured with this gyroscope effect (cinematic videos).

Maybe there are alternatives for my goal. I am open for suggestions :) :) :)

The Gimball is basically compatible. If the Osmo Action has USB C, charging the camera on the gimbal will also work.
As Danial says, the Gimball doesn't do much for you. I use it with the GoPro Hero 7 and can say that the stabilisation is better without the Gimball than with it. The small vibrations of the gimball motor are not defused very well by the software stabilisation.

For the DJI Osmo action camera, I would recommend the FeiYu Tech Tripod or the GoPro Shorty (tripod and handle).
Since the Osmo already has a super good stabiliser built in, you don't need a gimbal!