How are washing machine intake valve solenoids controlled?
It's difficult for me to imagine that these valves are truly intended to operate in the &quot;full bore&quot; and &quot;trickle&quot; modes you described. I think it's more likely that the &quot;trickle&quot; mode is actually a malfunction.
After reading a little about the differences in operation between direct-acting solenoid valves and pilot-operated solenoid valves at Burkert and at Instrumentation Tools, and also reading your linked question, I'll venture to guess that the valves in your machine are pilot-operated and that your water supply pressure is sometimes so low that the pilot operation fails. Maybe the water pressure vs spring is so low that the main valve opens only partially, or even not at all and the trickle is just what flows through the pilot.
The situation mentioned in that linked question, in which a flow-activated pump is installed to feed the washing machine, is interesting. But I think it's consistent with the theory above. As the pump is flow-activated, it does not provide any boost in pressure until after the valve has opened enough to allow sufficient flow. Thus the pump does not provide any additional pressure at the time the valve should transition from closed to open, and in fact probably is a pressure loss, further impairing the ability of the valve to open in the first place.