Hey PaintBallPaki,
First off, I think your choice in rifle will be good (
700 SPS Tactical AAC-SD Bolt Action) as it will work great as a deer/elk hunting rifle and the extra barrel weight will help soak up some recoil. Plus a short, stiff barrel is usually conducive to accuracy, which is always a good thing. The 20" barrel is threaded for a suppressor, which is an amazing thing to have on a firearm. There is nothing wrong with a suppressor on a hunting gun! Matter of fact, a lot of us would have been better off (
"What!? Ben's bitter about what?") if we had used one throughout our hunting careers. And until you get your suppressor, the gun will still work out just fine, 'as is'.
As for caliber, you've gotten some good suggestions but the ones I thought most thoughtful, based on your experience level, were for the .308 Win or the 7mm-08 Rem. They will both ethically take deer and elk, but without a lot of recoil. But the gun you want only comes in .308 Win and 6.5 Creedmoor, so that puts you into the .308 Win by default and that's a great choice. Lots of inexpensive practice ammo available and so many factory loaded options for hunting, it'll make you dizzy. This rifle and caliber are a great place to start your 'outside the blind' hunting future.
Here's a lead on one for $631:
https://grabagun.com/rem-700-sps-tact-aac-308win-20-hb.html
As you already discovered, if you really find your heart set on a different Rem 700 (or another rifle) that isn't threaded already for a suppressor, don't let that stop you from getting it. A good gunsmith will thread your barrel for $75 to $100 (depends on if you want the 'thread protector' with it or not. I would suggest you get it.)
Recoil is your enemy. Lack of available over the counter ammo is your enemy. The .308 Win is awesome on both fronts (moderate recoil & lots of ammo.) All of the magnums mentioned will surely kill elk. But you don't NEED them to kill elk. They will also kick you a lot in the process. And ammo will be much more expensive than the .308 Win.
As for the .338-06 A-Square, it is a SAAMI approved cartridge, but you surely aren't going to find ammo for it at WalMart. Down the road, when you have more time and experience and perhaps take up reloading, then perhaps revisit this cartridge idea. It is a wonderful elk/moose round. But when you touch one off, you'll definitely know it. And lots of recoil sucks the fun out of shooting and lots of shooting is what will give you skills and confidence to hit what you want.
I wish you the best with whatever you chose and welcome to the LRH forum.