Dong! (Sound of Marv chiming in)

Since the angle of swing does not affect the period, a clock pendulum doesn't need to swing very far from side to side.

It's a bit more subtle than that. If you write the differential equation for a simple pendulum, you can only get a simple solution (the one where the period is given by T=2*pi*sqrt(L/g)) if you make the assumption that the angle through which the pendulum swings is small enough that you can safely make the approximation sin(theta)=theta, where theta is the excursion angle.

Thus, if you keep the pendulum excursion small, it will act like a true sinusoidal oscillator, which is what you want. Since, in a clock, there's no need for large excursions in order to drive the escapement, designing for stable predictable performance is easier with small excursions.