WSWisconsin
Don't want to project that I am the subject matter expert, but I will say I have tried Gas checks with Paper Patching and, if they helped, I was not astute enough to see it.
It would appear to me that a Gas Check is just another variable that needs to be controlled. Hard to do, in view of the fact they weigh out a bit differently. Not much, but some differences. I discovered long ago that the base of a bullet is much more important than it's nose end, excepting very long ranges. With that though in the back of my mind and observing my sprue cut a little closer, I have managed to reduce my groups considerable.
I also use a wad...with a small bean size roll of lube on top to form a "cookie" when I seat the bullet. I am experimenting with a newspaper wad over my "bean" before seating. haven't realized any great benefit from that yet but I'm told to "have patience, it's there". we'll see.
Don't know about some of the other things you're doing on a one on one basis. However, they all appear to deserve consideration as part of the load "work up". I can say, with a bit of experience, that your paper lube should, and will, work. It is the same lube I've pretty much settled on after several years of pure frustration.
Respectfully, Russ