The pressure switch was not tripping simply because the tub wasn't filling quickly enough.
After removing the top (using a nut driver), I attached a different hose from the original inlet valve that drained into a bucket in order to verify that there was a problem with the valve. There was only a small trickle, so I ordered the replacement valve.
When this replacement valve arrived, I removed the original valve (requires a Torx driver, a T-20, I believe), put the new one in, reattached all of the hoses, tested, and then replaced the top.
Now that it's complete I can also say that the problem can also be diagnosed by listening to the machine when a cycle starts. If you hear kind of a faint trickle when the valve turns opens, the valve probably needs to be replaced. Normally, it's more of a very audible "whoosh" with normal water pressure.