You didn't say which language you're using. If you're interested in Rust, I wrote a crate which deals with a bunch of bezier operations, including boolean path operations: https://docs.rs/flo_curves/latest/flo_curves/bezier/path/ind...

The 'better way' you're talking about here is the curve clipping algorithm, my version of which is here: https://docs.rs/flo_curves/latest/flo_curves/bezier/fn.curve...

Finding the intersections between individual curves is only the first part of this operation: you also need a way to determine which edges are on the outside of the new path (flo_curves uses raycasting for this, same operation that the OP focuses on, essentially) and deal with a fairly large pile of edge cases - literally edge cases here. Things like overlapping edges, nearly overlapping edges, what happens if a ray passes through an intersection point or directly across a straight edge, precision issues, curves with loops, etc.