Updating this thread: The problem recurred, except the machine got stuck on a different part of the cycle, agitating my clothes for a couple hours instead of just a few minutes. The lint from that dryer load was insane. The brand-new shirt that was put into that load now looks like it's ten years old and well-worn-in. It's very soft, though.

Whirlpool repaced the timer assembly under warranty, no questions asked. I even placed the warranty repair order online, no time spent on the phone explaining it to anyone.

Odd that it's the second brand-new Whirlpool appliance I've had fail on me in the last couple of months...