Last time I found a paper in Graphics Gems titled Solving the Nearest-Point-on-Curve Problem, which transforms the problem into a Bernstein polynomial form. Then an exact solution can be obtained using A Bézier Curve-Based Root-Finder. This is my implementation [1], but it's not very robust for high-degree cases.

[1] https://github.com/Long0x0/distance-to-bezier

Your link 404s- private repo?