If recoil is a legit problem due to medical issues i would take 7-08 off the table—its +\- the same thing as a 6.5cm just with 150-200fps more velocity, which is not functionally relevant inside 300 yards. Of those two a 6.5cm would be my choice, and has plenty of factory ammo options, a LOT more than 7-08 does.
Hard to argue with a 243 shooting heavier ballistic tips, sst’s or a bullet like that, and seems like a proven solution for what you want. If you’re worried about a 243 being enough gun, then make sure to get a faster barrel twist so you can shoot slightly heavier bullets, and that should make up the difference. 6cm less common but would do the exact same thing. Ive never played with a 6arc but on paper it seems picture perfect for what you described.
Given your criteria 6.5 creedmoor beats the 7mm08, less recoil and way more available factory ammo.
I'm not sure if the get along with the no AR cartridge specification but the 6.5 Grendel 6 arc are well worth considering, a howa mini action in a Boyd's stock is a lot of deer killing with very little recoil. The 7.62x39 is pretty useful inside 300 yards too though if I'm stuck with 123ish grains in a factory load I do prefer the Grendel.
I think this chart of rifle cartridge recoil weights could be worth a look. Not a cut and dry chart on all the factors going into calculating the numbers, but it’s a good general guide to recoil weights
My opinion on the lightest recoiling rifle in my safe that I would personally be comfortable using as a deer rifle would be my 6.5 Grendel. Plenty of good bullet options available and good results out to 300 yards, which is in the wheel house of most hunters.
Being that we are on rokslide, it will only be about a year before guys here start shooting a .17hmr at deer as long as it’s as tikka in a rokstok.
I think the 6ARC is a great choice unless you believe it to be a niche cartridge. I personally think it is. The easy choice is a 243 and would be perfect. Couple its reduced recoil with a lighter bullet (80gr TTSX) and it will be joy to shoot AND do everything you want it to do. There is no down side. Get a rifle that has a threaded barrel because you will want to suppress it eventually. Once you shoot a rifle with a suppressor, you will get past its hideous looks.
All across the country, if someone was recoil shy, the 243 was the obvious choice and it still is a great way to go. There’s also reduced recoil loads that make it even easier on the shoulder.