A friend brought over an HP ScanJet 3570C flatbed scanner to see if I could fix it. It's dead, Jim.
Turns out the fuse is blown. Attached is a diagram of the power circuit. There's an 18R high-watt resistor in series after the fuse. That resistor is good, and the 7805 regulator is good. With the regulator removed, the output through-hole isn't shorted to ground. And the AC adapter is good.
So my first question is - with the resistor in there, how does this fuse get blown? 12V divided by 18 ohms is 667mA even with a dead short to ground. How does a 1A fuse blow?
Second, I don't have a replacement axial 10mm 1A fuse, or know where you would get one locally with RS being gone. I do have a large slow-blow (313) 1A fuse, but I would have to solder wires to the ends. Can you do that to a fuse without melting whatever is supposed to melt when it's supposed to blow? Given that the AC adapter must have some limiting built in, and with the 18R resistor there, would it make sense to just wire across the blown fuse at least to see if the thing is going to work at all? If it's still dead, then there's nothing more to be done anyway (it's all tiny SMD stuff, and I don't have a schematic). But if it works, then I could order a replacement fuse.