Are you able to mark the 100% closed throttle lever position with a pencil, or something like that, to see what happens when you have to push it to get the idle back to normal? This is one issue if it is physically sticking open, and another one if the throttle is actually closing but something else is keeping the idle up, until you touch it. I don't see how the running versus not running engine will cause the throttle to get physically stuck, assuming the same temp. etc. 🙄
Also, if you had the carb apart, did you remove the throttle from the shaft? I, once, had it not installed back very precisely, causing improper closing.
Moreover, I once saw a DIY accelerator pump repair, inside a carb, which caused the throttle not to close down 100%, but it was like this all the time, not intermittent.