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Yeah, seems like an ignition issue at this point. Even if it was too lean now you should have at least gotten an afterfire or two (which is not the same as a backfire).

I'm currently battling an apparent ignition issue on mine too. Runs great for about 15 minutes then one or two cylinders drop out. I initially diagnosed with an inline neon bulb tester which showed #2/#3 cutting out but after replacing the pickups all four ignition leads seem to be firing well but it's still cutting out. About the only thing left is to replace the spark plug caps; everything else has been swapped out. Coils, igniter unit, pickups, wires and plugs. Love intermittant problems...

I installed new Champion 801 spark plugs after gapping them. Brought it up with a bump of starter fluid. 37.5 pilots are too lean. Had to turn them 3.5 turns out to get it to idle. 125 mains (once you install the needles correctly) are too lean. It runs out of breath at about 4,500rpms. Installed the 42.5s, installed the 135 mains. 2.5 turns on the pilot screws. Coughs, won't start. How can something that idles like the vid below and has the throttle response shown below be so difficult to start? I'm convinced the enrichner circuit is working properly. At 1K RPM idle, pulling the enricher brings it straight up to or above 3K rpms.

I took the carbs off again. This is one time where I like pods, so much easier to deal with carb removal than the boots. I took the brackets off and separated the carbs. Removed the brass choke plungers, the bowls, stopped the caps, springs and sliders out of them. Put in new 130 main jets since the 125s were a bit lean according to Colortune.
I shot carb cleaner through the choke channel on each bowl, I shot carb cleaner from the brass choke plunger outlet through the brass choke pipe feed: shot out of there like a rocket. Nice full stream. They are wide open. Assembled the carbs, mounted them, hooked u the baby bottle, put the Colortune in cylinder #2. Turned on the fuel, tried to start without the coke, black as night. No fire. Pulled the choke out 35%, black, no fire. Full choke, black no fire. Brief shot of starter fluid, came right up. It chugs a bit at first, but if I remove any choke it dies. Very light throttle input helps it stabilize running.
Interesting to me, once at full temperature if I pull the choke it goes so rich it will kill the engine. And yet I can't get enough fuel to get it to light.
I've installed the new Dyna 3ohm coils tonight. I'll do another cold start tomorrow afternoon after I install the new wires. . I have some morning commits.

And it restarts without ether once it's warmed up?

Correct. Pablo on KZ650 suspects the choke jet is too small for the low vacuum and high air volume with the open air cleaner. He flipped a link to a tuner that walked through using a vise drill process to open up the jet. Certainly explains the symptoms.

Guess who just cold started on her own. Dyna coils and new plug wires, second bump it came up. I had good voltage to the coils before the change, coils and 3 of the plug wires all tested good. Swapped the bad wire with a tested good wire. Still wouldn't start. New coils and wires, boom!
I suspect the 2/3 ignitor is marginal as I see some misses on those two cylinders throughout the rpm range. The end is in sight!

Tried to cold start again today while watching the Colortune. Chugged and tried to start on the first try. Third try is chugged for about 7 or 8 seconds (blue on the Colortune) One quick hit of starter fluid and it took off. I've got to open up that choke circuit. Pin vise/drill arrive tomorrow afternoon.

What's the plan if that doesn't help? It's a permanent change with no going back, right?

I don't have a good feel for how sensitive these igniter modules are to bus voltage. I'm wondering if they drop off a lot in performance when it drops below, say, 10v as when the starter cranks. Might be a fun experiment to try and push start yours when it is cold then see if that makes any difference.

No going back, correct. I'm planning on opening it up a couple hundredths and see how it reacts. With the choke on I should see yellow in the Colortune when I get ignition, not blue. That suggests it is not rich enough for cold start. Opening up the choke jet a small amount shouldn't render the carbs useless. If it turns out to be too much then I just wouldn't open the choke all the way up. I'm calling it a choke, i should be calling it an enricher circuit.

As for push starting, how does that eliminate your concern about a potential voltage drop?

Push starting avoids running the starter motor, so the battery bus voltage would stay much higher than if the starter is cranking. If you've got a weak spark from the voltage drop then it makes sense why it will start on the ether and not the gasoline. Once it catches, the alternator is putting out enough current to raise the battery bus voltage back up and the spark has enough energy to ignite the gasoline. Then once you disengage the starter then the voltage goes even higher, so you are out of the woods at that point.

I think at one point you said that the plugs were wet but it wouldn't catch? I think my theory is consistent with that. I think it's also consistent with why it starts back up when warm: at that point the gas vaporizes from the engine heat and takes less spark energy to ignite.

That said, I fully admit it could be complete BS. I will in no way be offended if you decide not to try it and drill the orifices instead. Either way I've got my fingers crossed for you on this.

I always want to know the logic behind the action. Thanks for the lining it out for me. I haven't got the tank and seat on it but that is easy enough to fix.

FYI - One spark plug came out wet, (#1) the other 3 were just a bit blackened and sooty.

I removed the junction box, cleaned the corrosion off of a few connections, went through the start circuit and stop start button connections. Reassembled, voltage back in place. Not sure exactly what I touched but I'm back in business. I wrapped a hand towel around the K&N filters to reduce airflow and create more restriction. Tried to start from cold, not even a cough. The engine spins exceptionally fast with the new lithium battery. Still no joy. I placed some carb cleaning wires in the choke jet to identify the size and came up with .0185 inches. I used a .027 inch pin drill and opened up the jets. Grease on the tip of the bit, twist for a while, clean off the old grease and repeat. Tedious to say the least. Blasted a large amount of choke cleaner from the enricher plunger location up top back out through the carb jet to clear any drilling chips. The blasted from the brass tube up through the top and through the port from the carb bore into the plunger area. All ports, all directions and carb and choke cleaner liberally applied. This afternoon I'm headed back out to reassemble and try another cold start.