So, I guess this is not an electronic control washer, but has just a mechanical timer? Need the washer model number. The timer does not "command" agitation at the start of the wash cycle. In fact, during fill, there is not even power to the timer motor, so it does not advance at all. The water level switch is a single pole double throw switch. The switch either directs power to the water valves or to the timer and motor via timer contacts. When you set the timer dial to start the wash cycle, it is already set up to agitate but agitate doesn't start because there is no power to the motor via the timer until the pressure switch changes from the empty position to the full position. When in the full position, power is directed to the timer and is cut off from the water valves. Now the timer and timer motor have power and since the timer contacts are already set to start agitation, agitation begins immediately (before the timer even advances).

If the tub overfills before agitation begins, then the pressure switch is not working properly. If you had the lid open, then you have no way of knowing if the pressure switch activated properly without actively troubleshooting with a multimeter. With the lid closed, if it was just the water valve that was the problem, it wouldn't have overflowed until after agitation started.

Eric