But (approximating at small angles), the pendulum's period is constant. Friction will have the effect of reducing the distance that it swings, but the amount of time it takes to swing it doesn't change.

That's a simplified model of a pendulum and isn't completely accurate, but even if your pendulum has a 30° swing its true period is only 1.7% off of the approximation.

And pendulum clocks don't just let the pendulum lose energy to friction until they swing down and stop. Take a look at this deadbeat escapement: https://en.wikipedia.org/wiki/Anchor_escapement#/media/File:... (one of several pendulum mechanisms)

There are three main actions going on here. 1) A weight is hung so that its downward force is trying to drive the escape wheel's rotation. 2) The side face of the anchor (swinging part w/ pendulum attached) moves in front of the escape wheel tooth and stops it. This locks the motion of the clock mechanism to one tooth per swing (and is the reason the weight doesn't just drop immediately). 3) As the anchor swings off the dead face, the tooth applies a slight push to the sloped impulse face at the end of the anchor. That repeated nudge transfers potential energy from the weight to the pendulum swing, and keeps its swing distance consistent against friction losses.

Mechanical clocks are actually pretty clever.