I'm 6'1" and I don't have any problems shooting my Savage 12 LRPV which is a .223 rifle. Like others have said, there are add on recoil pads that effectively increase the LOP of any stock. They are custom built for many rifles. You wouldn't really need something to take the recoil of a .223 or a .243 but the length could help you. Personally I don't think you would have a problem with a rifle that has the normal LOP (13.5") with a recoil pad added (should increase the LOP but .5" to .75"). I have long arms in addition to being nearly as tall as you (and I'm not one of those people that add 2" to their height when asked about it
I can shoot 2" groups consistently at a quarter mile with my Savage. It came from the factory with a heavy barrel and a great stock (H-S Precision) and an even better trigger (a target AccuTrigger). Rifles like these are generally more accurate than the person pulling the trigger. I know mine is. I shoot off the shelf ammo too. Black Hills makes the best that I've found and it isn't all that expensive.
BTW there are plenty of people who shoot a .223 1000 yards and more. It's not as easy as doing it with a .308 but it certainly can be done. I've never had the chance to try personally since my range is limited to a quarter of a mile.
Mainly my point is that there are some very nice rifles around (made by several different companies). I like Savage and CZ (and I own both because I like them) because they have excellent triggers but so does Sako and Tikka. Sako is generally more expensive for the same amount of accuracy IMO but they are excellent rifles. I like Savage because you can get a great rifle at a decent price plus you can build your own rifle adding whatever parts you want too. For example changing out a barrel is about as easy as it gets on a Savage. Most guns require a gunsmith for that job but not Savage. Just a couple of tools and some quickly learned know how and you can DIY.
The thing that should determine what caliber you want is what you're going to do with the rifle. If you're shooting targets a .223 is fine out to at least 600 yards even for people who are just learning. For longer distances bigger calibers are better but that .223 can reach out there with more practice. If you plan to hunt what will you be hunting? Varmints are a lot easier to kill than an elk. So if you can give us a little more info it should help us help you with a caliber choice.