I would say the first 2 things you are going to see in the rifles in the 1300-2000 dollar price range is going to be the stock, trigger and longer barrel at equally light weight.
Most rifles in the $550-$1250 price range focus on the barrel an other features like threaded muzzles or soft touch stocks or cericoat. The name of the game for ties in this lower price range is out of the box MOA accuracy with most factory ammo. They also struggle to make a gun in this price range lightweight with out chopping the barrel below 24". There are some lightweight guns in this class but most show up with 20-22" barrels. There are a few exceptions to these rules however (Tikka T3X, Winchester XPR) but while both guns are light weight and have 24 or 26" barrels (and the Tikka has an awesome trigger) they both suffer from cheesey stock syndrome.
By stepping up to the 1500 or 1700 class of rifle you may find something with a very solid stock, long barrel, and good trigger. Often these guns come with a carbon fiber stock, and match grade trigger. The easiest way to put it is to say that while accuracy is attainable at the $550 price point, attaining accuracy at the $1700 price point CAN BE easier as there are less things you have to "fiddle with." Last January I bought a Tikka stainless hunter for $900. Beautiful gun but it shot federal ammo into 2.5". I remover the action from the wood stock and found that the recoil lug was loose and there wasn't really much of a good connection between the action and the stock. I made 2 aluminum pillars for it on my lathe and bedded the action and recoil lug with Brownells accraglass ($14). Now it shoots my hand loads into 3/8". I could have just bought a Christensen Mesa or ridge and avoided all that but I choose to save $800 and just go with the accraglass. I choose the wood stock because I planned on bedding it and I new the wood stock would be easier for me to work with.
Bottom line, if I didn't have a lathe or want to mess with bedding the action. If a gunsmith is going to charge you $300-$400 to "accurize" your $1000 gun and it will be at the shop for a month why not just buy a sako/Christensen/Bergara (premier) and have your nice accurate rifle NOW with no fiddling around with stocks or triggers.
Example1 the sako S20 right now has a rebate going where if you buy the S20 hunter you can send a copy of your receipt and revive a free S20 precision stock. My brother bought that gun and it's first group out of the box without break in was the size of a dime. The S20 is regularly $1600-$1700.
Example2 the Bergara premier mountain 2.0 at 6.2-6.4 lb in 6.5creedmoor or PRC it will do everything exactly that you want. Comes with a Bergara barrel, the gun is hand built in Lawrenceville Georgia and it features an AG composite carbon fiber stock and a trigger tech trigger. For the price of $1900-$2100 you could not build this gun for that price.
I would stay away from Kimber as there isn't really much value for the price, except you get basically a blueprint win m70 action.
The average savage 110/10/111 will absolutely do everything you want it to do I'd pick the storm model with the aluminum bedding block and 24" barrel (but that's like 7.5lb). Savage has an awesome trigger but their stocks are probably the cheesiest in the industry. The new models with the aluminum bedding block fix this at the price of weight.
Caliber.
Chosing a lighter caliber will definitely benefit and kill anything you want out to 500 yards. 308, 65creedmoor, 65prc, 25-06 7mm-08 and 270 are all great options. I know everyone hates the 65creedmoor but honestly chosing either 65creed or 308 gets you the most gun options because more guns are made in those calibers. So ammo will be easier to find than 7mm08, 25-06 and 65prc.
I know someone is going to get on here and try to say the 7mm 08 or 25-06 is more popular and why didn't I mention the 260 rem but honestly I go to alot of gunstores around the country and I haven't seen a gun on the shelf chambered in 7mm-08 since 2019. I have never seen a factory rifle sitting on the shelf chambered in 260rem and I did see one 25-06 but I've never found the ammo for it anywhere and this is from a guy who goes to enough gun stores to buy primers off the shelf through the pandemic. ....
Lots of other great options out there in the mid range:
Browning X bolt
Weatherby Vanguard