[old_tv_nut]

"Guess they figure since the supply line going to the diode has one of those capacitors across it, thing the supply side of it to ground isn't necessary?"
A cap across a diode and a cap from one end to somewhere else serve different purposes. One place doesn't sub for the other. There must be something different about the flyback transformer itself to require removing the cap.

Re clip leads: keep leads short and be very careful about accidental shorts, as the diodes would not like that.