I was able to get the Asteroids Deluxe game working with a little fiddling about -- I just reached up to where the fire button was and fiddled with it, and it started working, so it was probably just some dust or gunk between the contacts.
However, while the game works, there's a very odd and somewhat inexplicable problem. Or two problems. Or maybe four problems. I'd bet on it being two problems but it's hard to be sure.
The most puzzling thing is that the randomness of motion of the asteroids is usually missing entirely. They all start in the same spot in the upper right, and as you blow them up, the replacement asteroids stay on the same course as their parent asteroids, so you can clear the screen quickly by just plugging away at the cluster of bunched-up superimposed asteroids.
At the same time, the UFOs, when they arrive, have more accuracy than they should. It's nearly impossible to escape them. So while you never die to asteroid collisions you always die to UFO attacks, unless you get really lucky to have a shot in the air near where one appears already.
The "heart beat" sound also is missing. You still hear the ship thrusters and firing, the twirl sound of the arriving UFO, and the beeps of the arriving death star, but the quickening dum-dum sound is completely absent.
These three problems are probably one problem because they all happen together. Mysteriously, they all vanished for a little while yesterday and then came back together. It's hard to imagine what component could be failing to cause those three problems, but leave the rest of the gameplay working, though.
The fourth problem is that sometimes the screen's alignment is such that the top line, or perhaps all the edges, are not visible. But it's inconsistent, and not linked to the other three problems. It's probably just a matter of the CRT being old and physically failing, its beam alignment going out of whack at times.
I found a site which has the original operations and maintenance manual but had no luck finding any tips about these problems there. I'm at a loss for what else to try. There's nothing to 'reboot' with a power cycling, unfortunately; the only non-volatile RAM in the unit is only able to hold the top three scores (that's probably 16 bytes or less! but this was one of the first arcade games ever to have that feature), and resetting that doesn't help.
Still, if I can't get it working, that's no huge problem. The main point was to get the case. The game itself is just a bonus. It'll be a while before I can build a MAME machine, and it would be nice to have a game until then, but that's just gravy.