recoil
The only 6.5CM I've shot much is bamaboy's Ruger Predator, which is tricked with a Magpul stock and big scope. That dang thing will reach way out there! It's a Gen 1 and bottom of the price range or not, it will shoot with the best of them. Factory ammo to boot. My impression is that recoil is a bit more than .243, less than a 7mm-08. We've only shot steel with the rifle, but I have no doubt it would be grim death on deer, hogs or 'yotes at any reasonable range. If you look around, the Gen 1 Predators or the standard American models are still out there.
I have now watched and shot the .243 on deer enough to form a solid positive opinion of it as a whitetail rifle, despite criticism from many old timers when I was a kid. After watching my dad in his later years and bamaboy as a youth take whitetails cleanly, I started hunting one myself. Recoil is very soft, especially with the mid-range bullets in the 85 grain range. My 20" barreled carbine gets 85 gr Partitions when I can find them, but if your deer are not to big and you can pick your shots, the Sierra 85 gr BTHP might serve as a good dual purpose slug for deer and 'yotes. In my
Dad's 24" barreled sporter, I shoot 100 gr Partitions. Again, I'd lean towards a Gen 1 Predator, as I've got some experience with the rifles in other calibers and they all shoot well.
Bamaboy took his first two whitetails with a .223 and bonded bullets, and a pal started his grandson with a Mini-14 and I think 60 gr Partitions. It can be done, but I think there are better choices. I've got no experience with the 6.5 Grendel, but I like the numbers. I'm not sure if you can find a rifle so chambered in your price range. The .300 Black is not flat enough to suit me as a varmint-'yote cartridge, but I have a bolt rifle in 7.62x39mm that likely performs similarly and is a dandy woods rifle on deer. Again, whether or not you could find an affordable rifle in said chambering is another matter. The budget bolt rifles mentioned are available and offer more than the Grendel, the 'Black or the 7.62 Soviet can offer anyhow.