This library has a very interesting algorithm for computing the curve point closest to a given point, seemingly based on a root-finder that doesn't need any complex numbers. Does anyone know of any resources about such an algorithm?

The library only solves up to cubic equations, and the comments have a link to the following page: https://momentsingraphics.de/CubicRoots.html

For general polynomials, it matters a great deal in what basis it is represented. The typical monomial basis is usually not the best from a numerical standpoint. I am aware of some modern methods such as this: https://arxiv.org/pdf/1611.02435

For polynomials expressed in e.g. a Bernstein basis, there are often much faster and stable tailored methods working solving for the eigenvalues of a companion matrix of a different form.