Ugh... It is like fingernails on a chalkboard for me

One pair in my travel stuff always. Habit from grad school when I lived next door to a frat house

Great idea! Thanks. But maybe I like the opportunity to dance...

I would add tires to your list. Car tires, bike tires...

As an engineer, this warms my heart (and the resulting soup/rice mixture warms my tummy).

If you can make it go away with the shifter, I would guess it is not the limit screw but that cable tension (I don't think the shifter would help alleviate the problem if the limit screw is set too far in). You could start with using 1/4 turn increments of the barrel adjuster at the shifter. Go slow. Do one 1/4, shift around and stop back on large cog and see how it sounds and looks.