I'll have to say, I haven't read most of the intervening 5 pages. But I will agree, building a .223 AR for 1000 yards is a waste of good money if you want consistent hits. I say good money, because building for long range, whatever the caliber, should not be done on a tight budget. I don't agree, though, that the AR itself is not a suitable 1000 yard gun. The 6.5 Grendel, as a few people have mentioned, is quite capable at 1000 yards. I have an AR Grendel, and have hit targets at 1125 yards (18" square steel plates, so that's ~1.8 MOA at 1000 yards, about 1.5 at 1125, and I hit 6/10 at that range, and was doing 8/10 at 1000 yards. That's 123 AMAX, my loads, 2550 fps MV from a 20" barrel. I was making about the same percentage of hits as the three guys I was with, who were all shooting 178 grain SMK .308 Win; two from customized Rem. 700 bolt guns, one from a custom AR 10. I also consider them better shots than myself.
I can't argue that building a good bolt gun for the specific purpose of 1000 yd+ is probably a better idea, but if OP is set on an AR to build (actually just bolting parts together), a Grendel would be a viable setup. At 2550 fps, my rounds were still at 1255.9 fps @ 1000 yards, and were transonic at 1125. The drop at 1000 is 37.8 MOA (~378 inches, or around 31.5 feet), so if you do build something for 1000 yards, budget for a scope that has that much adjustment available, or add a 20 MOA mount. BTW, the energy of the 123 grain bullet at that range is about the same as a 165 grain .40S&W at the muzzle.