So this is still bugging me, since I've never seen one in my mind it sounds very difficult to fix. But if I replace that whole check assembly, in theory, if that is my issue, then that should rectify it?
To pull that tiny circlip that retains the screen requires tenacious dexterity that my mitts don't have. The carb I had saw some decent mileage. As well, it was equipped with the additional accelerator pump. I did all new diaphrams - including accelerator pump. Still the same symptoms. As Tim mentioned with the HS, sporadic rich/lean at WOT as well. Biggest consistent issue was the leaning out at idle or coming down off of WOT. I had previously tested the main check and it "seemed" to hold on vacuum. So, with that being the only thing left, I swapped the whole assembly. The capillary check is just a punched out circle of nitrile-like material - similar to diaphram material. But, it rests on (4?) bosses in the bottom of the pocket leading into the venturi. It had relaxed around those and wasn't laying flat like it should. I just threaded a machine screw into the brass housing and pulled it out. A new check and assembly presses back in and welch plug driven flush. The kit has everything including welch.
As you mentioned, if that were in fact the issue in the end, it is easy and inexpensive maintenance. I know the throttle shafts can wear out of round with lots of use, particularly with the heavy return springs on the Tillys. I've had similar symptoms when the throttle is released and the shaft shifts in the supporting bore, leading to intermittent lean/erratic idle issues. It all depends on the mileage on the carb and individual components.