Reaching out to 400-500 yards with a .223 is easy. Hitting something small at 400-500 yards with any rifle is a matter of training and the inherent accuracy of the rifle. If your rifle in the best of hands, from a perfect rest, is only capable of 2" at 100 yards, then it will at BEST, under perfect conditions, with the same flawless execution, be capable of 10" at 500 yards. This is NOT varmint accurate. So, in theory, if your rifle shoots 1/2" at 100 yards, it will print 2 1/2" at 500 yards, which is plenty accurate for even little prairie dogs. However, in the real world, there is wind and gravity, both of which are working against you hitting the target at 500 yards. Gravity, fortunately, is predictable, but wind is not. You can dial up any amount of elevation on your scope to have the bullet and the crosshairs meet at 500 yards, and you can dial in any amount of windage to account for wind, but all of these require knowing what you are doing. It's boringly routine, even in the hands of an amateur, to hit 1" groups with a 1/4" rifle at 100 yards. Stretch things out to 500 yards and unless you know how to read the wind and either hold to factor this in or make windage and elevation adjustments, you WILL miss every time. Small, light little bullets are even less forgiving of wind reading errors.
So, in short, yes, a .223 is more than adequate for 400-500 yards shots in varmints or targets, but it all comes down to knowing how to get there. A 300 MegaMagnum rifle will just be a more expensive miss than the little .223 if the operator can't do his part. All the expensive hardware and glass in the world will not make up for the deficit in the operator. You can see this play out every trip to the range with folks with very expensive rigs who have no business shooting at even deer sized game at any distance. Practice, practice, practice.
Also, don't waste your money on the new accustock. Buy an older Model 12 series and replace the stock. After some research, I found the best bang for the buck was the Bell and Carlson Medalist in tactical configuration. The lower end B&C stocks are better than the Savage stock, but not as good as the Medalist. If you really wanted to do it right, just buy a model 12 action, or a cheap used model 12, and replace the barrel with a high quality aftermarket barrel. Drop in a nice trigger for $150 from Rifle Basix and you're good to go. A relatively inexpensive rig to put together and it would be shocking to see it shoot anything worse than 1/2" groups in good hands.