Well, I could get the service manual for about $25 online for this model, so I may just do that.
I've managed to get it to turn on semi-consistently by briefly shorting two pins on the Analog Devices AD9884A chip. When I short them there's a high pitched buzzing that seems to be coming from the power supply, then I release the short and the signal light goes green and I can access the on-screen display. If I then do a Reset from the menu, it correctly locates the input signal and shows it. However, I still can't switch to other signals through the buttons, and if I try to use the +/- buttons to switch display, I can no longer even get into the on-screen menu.
Additionally, shorting those two pins doesn't work EVERY time, just regularly.
I suppose it probably is best just to buy the service manual, though I'm not sure it's going to offer any better suggestions.