Thanks for the help!
The relay isn't chattering, and the 24v seems to be all good.
In fact, I've discovered the problem... I took it all apart (nothing to lose) and found a burnt solder connection on the back (underside) of the circuit board.
I got to this point by poking around with a meter... it gets a good 120v on the black (hot) wire, and 0v on the orange (output to blower) wire. So, going back to where the wire comes out of the circuit board, still zero.
On the hot side of the relay (the contacts of which are clean and make firm contact), it has 120v.
Flip the board over, and the pin from the hot side of the relay is burnt. See the photo.
I am going to resolder this joint, which is easy enough. But I do wonder what caused this to happen... this furnace was installed in 1996, and this part has presumably never been replaced.
What could cause this to happen at this point? Serious over-current?
The high-pressure pump was pretty gunked up (seriously slimy screen, needed a dip in a carb-cleaner bucket even after being disassembled), and caused some noticeable drag on the fan/motor before I cleaned it. After cleaning, on the bench wired direct to 120v, it spins like a top, no vibration or noise.
So maybe the drag caused the 1/7hp blower motor to lock up, which caused a serious current draw (44 amps according to the label on the control) which burnt the solder?
We'll see if it all works when I get it back together in a few minutes...