I have had a number of AR-10s and a SCAR 17s, which is an expensive FN-made .308 used by our military, I have none currently, but do have an ATN X-Sight II.
Armalite invented the AR-15 and AR-10 and is a great way to go. You might find one on Gunbroker for 1300 to 1500, less if you were willing to buy a used earlier model. My two Armalites were totally reliable and well-made. You could also go with a Smith M&P10 new on Gunbroker for 1100 or 1200 with no iron sights. I'm sure they are very reliable and probably the best deal going. I would consider Smith to be the lowest you should go on a new rifle. And NEVER buy something somebody built in their basement. Hard to resell, who knows what's really inside. .308 is far more versatile than .260, which is a wonderful caliber. You have a wider range of loads in 308. Bulk ammo for the range is more readily available. .308 gives you the option of heavy loads, like 180 gr, for bigger game like elk. For coyote and whitetail, either caliber works great.
The ATN scope is really very slick for the money. It's great to be able to sight in during daylight without peep hole covers. The digital picture in daylight is great. In darkness, the black and white picture is more clear than the "color" picture, which is very green. Switching back-and-forth between day/night and B&W/color is very easy. If you accidentally switch to night during the day, it does not damage the scope. I have the 3-13X model,iirc. The higher range would make it difficult to acquire a close-in target. If they made a lower range, I would have chosen it. You should buy the external battery pack. The scope works as advertised, but don't expect Gen III night vision for 600 bucks.