Washer fill stops after a few seconds

I am running into trouble with a Kenmore clothes washer, model No 11026732500.
THE SYMPTOMS: the fill segment of all wash cycles opens the water valves and runs fine for anywhere from 10 seconds to perhaps 30 seconds and then it suddenly shuts off having not properly filled the tub. When it shuts off, there is no power to the timer unit – i.e. the timer motor is not operating as observed through the hole in the back of the timer case. Pushing the Timer Switch in and pulling it out again does restart the fill, but the same thing happens: fills for several seconds and then shuts off. I did observe one instance where the fill sequence did in fact operate until the water level switch shut it off and transferred power to the timer correctly completing the cycle, but once in 50 tries doesn’t wash clothes.
There is no water in the plastic tube to the pressure switch, though blowing through the tube to the tub produces bubbles in the tub water. Blowing lightly into the pressure switch connector produces a click and starts the agitator motor; releasing the pressure starts the flow of water, again only for the few short seconds and then it stops. On the chance that there may have been a diaphragm weakness or a slow pressure release, the pressure switch was replaced – to no avail. As well the water-mixing valve was replaced on the possibility of intermittent electrical problem therein– A relatively easy and inexpensive fix, but again to no avail.
QUESTION 1: A basic electrical schematic with the machine shows a Timer Switch #2 [FOLLOW BLACK LEAD DOWN LEFT SIDE OF SCHEMATIC] that is identified on the switch open/closed contact chart as a ‘bypass’ switch that might (??) provide current to the timer. Is this the case? Or is it a switch that is energized by the timer to power the water level switch further on in the circuit?

Just beyond Timer Switch 2 in the circuit and seemingly in parallel with one another is a pair of Timer Switches numbered 31 (extra rinse) and 10 (fill).
QUESTION 2: Timer Switch #31 seems to feed a separate component that is not always used on all models. This model does have a control board switch identified as Rinse 1 and 2. Is this Rinse 1 and 2 the “extra rinse” of which they speak?
Timer Switch #10 (fill) appears to supply the center (violent) lead of the water level switch that either opens the water mixing valves or powers the motor.

QUESTION 3 and perhaps the most important: Since I have no means of telling whether these switches (2 and 10) operate incorrectly at the time of the problem, from your experience, is it in the realm of possibility that the contact(s) of the Timer Switch 10 are bad (loose, burned, pitted, broken o/r breaking) so that they cannot /do not maintain contact to supply voltage to the water level switch and on to the water mixing valve? From your experience is it the case that a bad or failing timer switch set can cause the symptoms I described?

Obviously I would prefer not having to replace the timer, especially if it is only to find out that it is not a bad part. Therefore, any help or leads that you can give me would be most deeply appreciated.

Thank you for your time and help.
[FONT=&quot]BaldHeadedGuy[/FONT]
WasherSchematic6.pdf (544.2 KB)

let,s say it,s #3..by removing the time off the washer, it,s easy to take a part.Remove the plastic slide and check for burnt contact[s].If so take a emery board[sand paper] and clean that contact[S] then install and try…gluk

Well, it sounds as though it is relatively easy to examine and tinker with the timer. My concern was that the easily opened snap-latch box would likely contain 3 million small pieces under spring tension so they’d spread themselves far and wide with 3 or 4 of the more important parts hiding under the 3 year pile of ‘gotta keeps’ in the corner. Your answer suggests it’s not all that hard… crash helmet on, goggles down, charge!!!
Thanks for your help. We’ll see how it goes.