... And, what about the cathode to plate connections on this tube? Why not just parallel plate-to-plate and cathode-to-cathode? ...
At first I saw that and said, "Dafuq??"
You see where the ground connection is. Look at the other end of the tube, and follow the non-grounded plate/cathode to see where they go. They connect to the output of a tone circuit and a tube grid with no other ground reference.
So current can flow through the 12AU7 if the signal at that tube grid (6AN8 input) is positive-going, but would be blocked when it switches to negative-going on the other half-cycle. So the other 12AU7 triode has to have its cathode pointing to that 6AN8 input to allow current to flow on the negative half-cycles.
... Any special reason why they would use this over a typical oscillator setup? ...
They wanted to prove folks wrong about never seeing a 12AU7 in a guitar amp?
Maybe they got a kick-back from their 12AU7 supplier for follow-on tube sales?
Fender already had a patent on the neon-optoisolator trem and they wanted to stick it in Leo's eye?
Who knows. Try it and see if it sounds good. I probably won't be wasting a tube socket and heater current like this, but I do have a metric sh!t-ton of 12AU7's just begging for something to do...