That's odd about the agitator, not so much because it has a Kenmore transplant, but because it's the second WP I've seen with that same transplant in less than a week. I got a 1973 TOL Whirlpool from Arizona with a gold straight-vane in it recently (Anyone interested in that machine?)
In the case of this washer, it is very possible that the transmission has been replaced. To power that gold agitator, a splined agitator shaft is required. My earliest recollection of that being used was in the 1965 Lady Ks with the gold Roto-Flex. Everything else prior to that had been equipped with either the Surgilator, the 6-Vane (straight vane) and Roto-Swirl/Super Roto-Swirl, all in black bakelite. I don't know if machines prior to the 65s had splined agitator shafts for a couple years, but to my knowledge, machines prior to the development of the plastic agitators REQUIRED a drive block agitator and had a single-groove agitator shaft, and if these are 1964 models, they should have as well.
So, I'm guessing that the transmission was replaced at some point, OR if it already had splines, then at least the agitator was swapped.
NEAT washer though - love the pink. funny that I've already had several folks telling me to go after these machines. I do't plan to however.
Gordon