Hey Mike,
It depends. I'm at work, and terrible at making graphics, so I hope this noodling is clear (and correct, since I'm not an exprt).
This basically explains how the signal is routed through the switch:
Switch down, switch in the middle, switch up:
As you can see, the On/Off/On gives you 3 options: Switch down routes the signal through a diode configuration, switch in the middel just sends it straight through without diodes, and the switch up sends it through another diode configuration.
So to answer your question about the On/On switch: It it possible, but you would only have the 1st and 3rd option in my drawing. You have to choose 2 options out of the 3 there are now. Like no diodes in the bottom position and diodes in the top. Or only diodes, etc.
I'm not sure, but I think originally there were no diodes in the pedal?
If the above makes any sense, it should answer your question about omitting the switch. If you choose no diodes (the middle position) the switch doesn't do anything, it just passes the signal. So you can just see it as one wire, and connect the 2 as if there was no switch in between them.