See attachment. I hesitate to object Mr Penrose, but I think this exercise shows quite clearly the superiority of the Lagrangian method from a calculational point of view. If one tries to solve the plane pendulum using Newton's laws only, one has to speak about radial and tangential components of the gravitational force, the centripetal force etc. - whereas the Langrangian method yields almost immediately the correct answer (if one uses suitable coordinates - I have no idea why Mr Penrose suggests cartesian coordinates in his hint).

As you clearly realized, the best way to do this is to ignore Penrose's "hint", and take as your coordinate right from the beginning.

The Hamiltonian method is slighly more complex in this case only because you first need to find out what is.

The direct Newtonian calculation is much easier than you make it out to be - it's in fact the easiest method: Given that the pendulum length is fixed, you can ignore radial/centripetal forces altogether, and just calculate tangential acceleration due to (the tangiential component of) gravitational force: is the immediate result!