Just a thought
When you say you can't get the water to drain, is the washer attempting a spin cycle? These machines have reversing motors, so during agitation the motor turns one way, then during spin it turns the other way. The duration of the spin cycle is controlled by the timer, so if the motor isn't reversing the washer would appear to do nothing for a spell as the timer continued to advance until the spin time was used up. Then since the water hadn't drained out the fill level selector switch contacts would still be closed and the timer would continue on with the rinse cycle agitation.
The plastic Maytag pumps are pretty bullet proof so it's hard to imagine one seizing up unless some foreign object got inside and jammed it. Also noteworthy is that pump is in constant motion during agitation and spin. If it truly is seized up you would likely smell and hear something as the belt is dragged across the stationary pulley - that is unless the belt is missing altogether. I suppose that's also worth looking into.
[this post was last edited: 8/15/2016-02:22]