Use After Effects or some other video editing software to create a video. Then there are many tools to create gifs from the video. I use ImageMagick myself.
To make the image smaller I installed and used a command line optimization program called gifsicle. The command line can seem intimidating, but it's really not, just another process to learn.
I tired my usual tricks in PS, and nothing short of making the image size 400px wide and changing the dither type worked well. This method only required me to reduce to colors (which you can see the effect of in the first part on his neck) but overall I was surprised at how much quality it retained.
5/5, would use gifsicle again.
Seconding what doreadthis said: you'll generally need 2 programs, one targeted towards video editing or rendering effects (such as Resolve or Blender), and another targeted to compiling images or converting video to gif (such as Gimp, Gifski, or https://ezgif.com/)