Long range shooting is an expensive hobby
I have hits out to 1400 with my Larue OBR, with 16.1" barrel in 7.62x51, on IPSC steels, using M118LR, and the help of an excellent spotter. The mechanics of shooting at distance aren't hard. What is hard is being good at calling wind.
At the last school I went to, they were making hits on the same targets at 1200 yds with 77gr Mk262, out of 16" AR15's. No idea how much oomph was left on the round at that range, but the hits were consistent, and I wouldn't volunteer to catch one of them.
To answer your question, the AR's you'll be able to be sure will get you accuracy are not cheap. You might luck into one - The guys on my PD's sniper element all bought personal DPMS AR's in .308. Of those, all but two sold them, because only the two would consistently hold better than 2 MOA. One of the guys in the school I went to started with a DPMS with a Counter Sniper brand scope. Day Two, the scope launched springs and small pieces out of the top turret, Day Three, the DPMS started failing to reset, and shooting inconsistent groups, no matter who shot it. He finished the last 7 days with a school rifle.
I know guys who have built extremely accurate .308 AR's - some were very reliable, some require constant tweaking. The "standard" in AR10 parts varies even more wildly than that of AR15's. You might get lucky, and everything works great together, or you can tolerance stack yourself into a real s-storm.
Off the shelf options that are consistently solid are Nemo, Larue, GAP, and repotedly, the newer KAC. They all come with a stiff price tag.
Shooting a semi accurately takes more concentration than shooting a boltgun accurately. The boltgun will always be more forgiving of minor screw ups on your part. If you can consistently do your part, modern semis can do some crazy stuff at distance.