Jim -
I was going to comment on that machine also. This model or one VERY similar was available into 1973, in fact it inherited as a model revision the 1972 lid (notched with no separate handle) before it was finally discontinued. There was even a 1974 large capacity version that differed only in the color of the panel inset.
As to what was wrong with your machine - sounds very much like the springs inside the transmission (there are two - one on the agitator shaft and one on the gear-fork shaft that pulls the transmission in and out of 'gear') had worn out or broken. After many repeated switches in and out of gear, these springs tend to fatigue, and are generally a standard replacement item when a transmission is opened. If a transmission has been water-logged, which happens especially in later machines, these springs actually can rust and break.
When the spring(s) fail, the agitator shaft no longer has the downward pressure on it needed to slap into 'drive', and the agitator will sit there and at times fail to fully engage. Pushing down on the agitator will often make it go into gear fully, or enough to remain there. A replacement gearcase or a transmission rebuild would have been necessary to fix your Mom's washer.
Gordon