I think one can simply use the Euler-Lagrange method which is able to account for the constraint forces acting on the ball. Haven't worked that out for this particular problem but it should be relatively easy. Davies argument is a bit overcomplicated I think, the main challenge here is correctly accounting for the geometric constraints in the movement of the particle. I find the argument about the higher-order derivatives a bit weird as well, the system can be fully described using its potential and kinetic energy which are scalar (possibly time-dependent) fields and implicitly contain all forces, given some initial conditions (position and momentum) we can solve the equation of motion of the system with that.