Well, I agree with you that the problem seems to be in the idle circuit. If that's true, you may need to remove the mixture screw to thoroughly clean that passage. It is a PITA but sometimes is very necessary.
You will need to destroy the anti tamper cap to access it. That cap is glued on with blue Loc-Tite. Some guys just pry them off but that breaks off the false head of the mixtures screw. I like to use heat to melt the glue and leave the head so I can use it to help get the screw backed out.
Look at the low speed emulsion tube (jet-set) VERY carefully...item 18 in the link. Those things can be impossible to clean and, when I was working on these things every day, I didn't even try. I would just replace them.
The little oring at the top of that tube is CRITICAL. I never reuse them and I apply Vaseline to them before installing the tube.
The vertical orifice in that tube is so difficult to clear it wasn't worth my time to screw around with it.
Also, those tubes will crack vertically. They are a pain.
While you have the tube out, spray cleaner into the port it attaches to and observe the idle orifice in the carb throat for flow.
Prop the throttle wide open so that you can see every delivery orifice....idle....transition and main/high speed.
Use the spray straw on the carb cleaner and cover each of the delivery orifices with the tip and "backflush" each one. In this case, pay special attention to the idle.
Locate the accelerator pump delivery tube at the front of the carb throat. Place the cleaner straw over the spray orifice by inserting through the carb throat from the throttle plate end. Make sure fluid comes out the pump delivery passage.
Spray into the accelerator pump delivery passage and VERIFY that fluid sprays from the accelerator pump tube orifice and out the back of the carb throat 8 feet.....that's right, 8 feet.
If that stream isn't 8 feet, keep spraying into the orifice to "backflush" and then into the passage entrance. Back and forth, back and forth until that orifice, brass tube and passage is open and delivering an 8 foot stream out the back of the carb.
If you don't do that, you're likely to have any number of symptoms with the way the engine operates.
There are two passages in the bottom of the float bowl. One feeds the accelerator pump piston and the other one feeds the SE circuit supply "well". Those can get packed with "mud" and you might need to use wire to clean them out.
Lastly, there are two passage entrances on the front carb flange. One is for low speed air jet and the other is main air jet.
Hose them out with generous amounts of carb spray.
It's a good idea to reverse flush everything.
Use your fingers to block off passages to increase the power of the spray.
Make sure your bowl air vents are open.
Just a couple of tips from an old man.
Good luck.