Good point. I think, that if you're aiming for the speed, Chebyshev polynomials may even be excessive. It's unlikely that a high order polynomial will beat the hardware implementation. And for tiny polynomials it's best to spend the sparse grid values for the special points.

E. g., you can get your approximation both smooth and continuous by adding the target derivatives in +-PI/2 into the Vandermonde matrix. The overall error may become higher with the same polynomial order but the approximation will be smooth.