I've been really happy with my Sako chambered in .260 in the caliber range you mentioned OP. I've taken some real long shots on appropriate sized game with it and have been lucky to be very successful. I've also used it in some long range varmint shooting scenarios and was happy to punch a jack rabbit at 823 yards. Good thing he didn't know he was being shot at on the first try
I'm a big believer in "having the right tool for the job" so a "do it all rifle" doesn't really exist in my hunting world. I suppose if I had to empty out the safes and only keep 1 rig I think I'd keep my 28 Nosler.
I shoot a little 162 Grain 7mm (.284) pill that still has 2,200 ft lbs of energy at
500 yards. In comparison, that's 100 ft lbs
more energy than a 180 grain .300 Win Mag has at
350 yards! Also, a .300 Win Mag shooting a 200 grain pill out of the same rifle has 2,100 ft lbs of energy at 500 yards. The little 162 grainer still beats it by 100 ft lbs, and obliterates it out past 750 yards. Shoots flatter, hits harder, all with less felt recoil.
Do you need something like that to be successful? Absolutely not. I've killed more animals with my Dad's Winchester Model 61 than all my other rigs combined. That thing has probably killed 1,000 snakes, 3,000 squirrels, 1,000 rabbits, and put plenty of holes in hundreds of desert dogs down in southwest Arizona.
I've killed more deer with an old 18" short barrel Remington Model 742 than any other gun... That doesn't take away from what rifles and calibers are capable of today. Use whatever tool is available to help you be as successful as possible. For me, right now, and how and where I hunt, I'd roll with the 28 Nosler. Figure out what rifle that is for you and buy it and shoot it and enjoy it!