Hah! I found the problem. Not at all what I suspected.
I blew out all passages and jets, but no improvement. Then I got it running as well as I could, still popping and not running evenly. I then started pulling spark plug leads. The engine idled with the #1 and #2 cylinder leads pulled but quit when either #3 or #4 were. That cause a light bulb to go on, so I checked the balance of the two carbs. Bingo - #1 and #2 were pulling 5 and #3 and #4 were pulling 7. Got them balanced and the problem was fixed.
So what was happening was that the butterfly valves on #1 and #2 were almost completely closed. The two aren't exactly aligned, so #2 was pretty much completely closed and not pulling enough vacuum through the idle adjustment screw to make it effective.
The lesson I learned is not to mess with the carbs without checking the balance first!