This sounds like something seen repeatedly at virtually any bike shop every spring. It may not be the case in your situation, but the usual culprit is when gas is left in the carbs over the winter. It dries out and leaves residue in the float bowl. When gas gets to that residue it flakes off and turns into gummy stuff. It usually plugs the pilot/idle mixture circuit. In other words your pilot circuit, aka idle mixture circuit, is plugged up with dirt or gunk. It won't be easy to just blow out with an air gun. It requires more than that in many cases.
Don't root around wit wires in all the holes, scratching up the surfaces affects fluid flow. I don't know how it would work, but maybe some heavy gage fishing monofilament might just work well to poke and prod around in the circuits since it can bend easier than wire. Maybe some 20 or 40 lb test line would work. Never really thought about that till now.
The thing is the pilot jet and the circuits through the carb are tiny and can easily plug with any sort of residue. If you pull out a pilot screw and compare the hole size with that of a main jet you will see what I mean. Makes the eye of a needle look huge. It is possible you might even find one or both pilot jets are clogged when doing this. After all that is the smallest part of the passages. If clogged, clean very carefully with a pin and cleaner. A carb washing machine is the best way to clean out the circuits in a carb body set, letting the carb bodies slosh around for an hour or two. Otherwise try to fill the pilot jet hole and circuit with cleaner and finally with water and air. Yes, wash them out under warm water, then blow them out.
In addition, there is one idle mixture screw per carburetor and I'm thinking they're at the back of the float bowl on the bottom between the float bowl and the intake, but it's been so long I don't remember for sure. There is a screw on the linkage for balancing the carbs and a large screw that adjusts overall idle for rpm setting also on the throttle linkage, don't confuse these with the idle mixture screws. The idle mixture screws may be capped off if they were controlled by EPA back then. I'm thinking they were. If so, a little heat from a soldering gun will melt the glue if the cap is over a protruding screw and it can be pulled off. That or grind off the stops if possible. On some bikes the screws are in a recessed hole and the cap has to be drilled and pulled out.
After making sure all is clean in the pilot circuit try again.
KLX650C, Zephyr 550
SR500, Bultaco Sherpa T