I'm not sure how much that article helps. It only talks about how to combine two cubic Bezier curves in such a way as to be smooth.

Anyway, Wikipedia alludes to the solution: You bisect the curve and check if the result is flat enough to be approximated by a straight line, or round enough to be approximated by a circle segment. If not you bisect the part you can't draw outright again. Rasterizing a straight line can be done with Bresenham's algorithm, and there is a circle variant of it, so I won't dwell on it. You can Google if you want. The question is: How do you calculate the flatness criterion? I don't know, but when I search for "Bezier curve flatness" online, I find this article:

https://www.joshondesign.com/2018/07/11/bezier-curves. And it talks exactly about drawing Bezier curves by piecewise linear approximation. So exactly what you want to do. OK, it is in Javascript and you need it in C, but that hurdle can be overcome, right?