The bad news is that I no longer have a solder-sucker or any desoldering wick, so I wasn't able to do the larger part of the Asteroids repair.
In case you don't know what I'm talking about: the repair involves removing a bunch of capacitors and transistors from a circuit board, which are soldered in, and then soldering in replacements, to "reinvigorate" the 25-year-old power supply. The idea is that a bad power supply can cause other components to misbehave or fail, so replacing $10 worth of capacitors can keep the rest of the game working fine. But removing components from a circuit board when they're soldered in requires two steps: melting the solder, which any soldering iron can do, and then removing that solder while it's still melted, which is where it gets tricky. One way is to use a wick that soaks up the solder; the other is a pump that sucks up the solder (possibly built into a special soldering iron just for desoldering). I have neither, apparently; I must have lost the pump I had in a move or something.
The good news is the other two parts of the repair, though much smaller, I was able to do. I replaced the "Big Blue" capacitor (a capacitor almost as big as a can of soda, that forms a backbone of the power supply), and I also replaced the "Pokey" chip from the circuit board. (That's the big chip in the middle.)
And the really good news: that fixed it. The game works brilliantly now. The Pokey chip is what did it: that's what the diagnostic had shown as failed, and that's the chip that contributes the sound and randomness which were missing. The other stuff was primarily preventative, and secondarily intended to fix problems that couldn't be easily diagnosed and fixed.
I failed to buy a solder-sucker pump today at Home Depot (they have very little electrical soldering stuff, mostly plumbing soldering), so I'll have to stop at Radio Shack some time to pick one up. Then I can do that preventative power supply rebuild anyway, just to keep the game in good shape.
At this point I'm leaning towards not converting it to a MAME system, but preserving it in its current form, and building a separate MAME cabinet. The difficulty of adapting the Asteroids case to fitting a full MAME control panel is probably not worth it.