Austin, the 1957 Frigidaire Imperial had a very unusual thermostatic water valve, it was only used for one year on the WI-57 and never used before or after that again. The 1957 valve cycled the hot and cold water on and off to mix the warm water into the wash tub. So for example the 1957 valve would will fill for 30 seconds of hot, and then fill for 35 seconds of cold or what ever amount of time its "brain" decided was necessary to make 100 degree wash water.
All the other thermostatic valves in early and later Frigidaire washers as well as most other brands too mixed the hot and cold water right in the water valve (just like the valve I have pictured above) so mixed warm water was directed out of the fill flume onto the clothes. The drawback with the 1957 style valve is it would direct hot water directly onto the clothes. I have no idea why Frigidaire decided to experiment with this cycling style valve, but it was quickly discontinued for the 1958 line. Interesting to note that 46 years of design later, my 2003 Harmony washer does something similar to the seperate cycling hot and cold water.
Hi Roto, opening the "bad mechanism" is low priority, I want to get this machine up and washing first. Hopefully I will be able to open the mechanism with the missing pulsator shaft soon.