You can turn the curves into polygons, or render them to textures. Ruffle, I recall, makes everything polygons (so it's a little chunky if you zoom in?), and Super Meat Boy rendered everything to textures.

I'm not sure what the actual flash player did, which seems to have pretty decent performance relative to Ruffle in my testing.

Maybe they have some proprietary technique for rendering curves quickly on a GPU? (I read a paper on rendering curves, and there's OpenVG, which I think came later and nobody uses?)