FWIW, I have a 30tooth rear cog and the little wheels in the derailleur DO "mesh" with the cog.

BIG hint: I added a stem riser and taller handlebars which effectively "stretches" the shifting cables. I already replaced the brake cables and housings because they were too short.

I SUSPECT, as others have mentioned, this problem is one of too short shifter cables/housings. (at least in my case)