I've been experimenting some more and things are getting even more confusing. What I thought was a stuck float appears to be something else. I removed the fuel line from the offending carb and blew into it, only to find that the float needle appears to be seating OK.
I put the fuel line back on, then turned on the key and looked down inside with a flashlight. I can hear the pump run for a second or two, but everything looks OK. Then when I start the bike, fuel appears to start leaking around the jet needle. But if I open the piston manually while its running (without opening the throttle valve), the leak stops. As the bike continues to run, I can see fuel creeping up the main jet air bleed, next to PAJ1. If I turn off the key, fuel sprays out of the main jet air bleed several feet in the air. If I allow the bike to continue to run, fuel eventually reaches the top of the main jet air bleed and starts to overflow into the throat. The bike then starts to smoke out of the offending cylinder, I can hear that cylinder start to foul, and ultimately cut out completely.
An unrelated observation: while all of this is happening, on a different carb, smoke is pouring out of the hole which is adjacent to, and counterclockwise from the main jet air bleed hole. The smoke continues for several minutes after I shut off the engine.