Ho ho ho. Not all moons truly orbit around their planet, it depends on their relative masses. They orbit around the centre of mass. Which is never in the centre of the larger body (eg Earth) and in some cases the focus is outside the larger body, both orbit around nothing. Oooer.

One thing the video shows, it that the JWST's trajectory is not normal to the plane of the Earth but at 33º off 90.

And the narrator does mention that all the planets affect the orbit (and moons). Adding to normal corrections every 21 days to maintain an elliptical orbit. I actually got into the video after the first 30 seconds.