Cool, now at least I can replace the switch. So, I poked around some. The power supply is on the same board as the clock. And it is working just fine. Even though I don't see how the power supply not functioning could cause something to be always on, I tested all the resistors, and swapped out the e-caps. I even tried substituting in different diodes, it made no difference.
Now it looks to me like where the B+ hooks into the radio board is connected right into the power supply, so it would have to be on all the time. The switch hooks into the same spot on the power supply as the radio, while what I would consider the output of the switches (on/off and auto/manual are tied together) seems to go back into the clock circuits. I can hook the what I identified as the main B+ input to the radio board to that 'switch output' and the switches function like you would expect and turn the radio off. But the alarm (or sleep function) does not work when it is connected like that. It doesn't look like the wiring was ever modified, so I'm sure I just don't completely see how it is supposed to work.
Then I shorted something out poking around, and now the radio is dead (even though there is power getting to the radio board)
I don't think I'm going to get this to work without either a schematic, a 2nd similar Yorx clock radio for parts, or both. I almost never work on anything this modern, but that Realistic clock radio was such an easy unit to repair (with just those 2 bad coupling caps in the left channel and the dial light out) that I thought I would try another.