Oooh the colour combo they used on the video would make a gorgeous lionfish custom!
Enamel paints cannot be used on ponies, I know that much. So if those were used on the ponies Einhornbaby mentioned, that could explain why they turned sticky. Enamel paints are great for any hard plastic items though. Perhaps some pony merch would be suitable for this purpose, IDK, McD ponies or such? Anything that is hard plastic rather than soft vinyl like the regular brushable ponies.
I wonder if the marbling technique could be done using acrylic paints instead? I did some googling and found
this tutorial for kids, perhaps it could be tried on ponies? This "recipe" uses corn starch as the thickener. I'd imagine airbrush acrylics work best for this purpose, as they are quite thin to begin with. A primer, like recommended on the video in the OP, sounds like a good idea. But I always recommend that for any full body repaints anyway, as it gives the paint something to stick to. If anyone decides to try this, I'd love to know how it goes!