Dont overlook a Savage rifle. Out of the box I have seen quite a few that would run circles around a comparable Remington.
Typically they are a bit cheaper to accurize.

As far as buying vs building it all depends on how far you want to go. You can pick up a used 10FP or 700 for about 5 bills if you look. Then you can take your time and only make the changes you want as you can afford them.
If you are just starting in the bolt gun/distance game it will take you a loooooong time to really see the benefits of a tricked out build.
You are going to be a bigger variable than a stock vs built rifle at first.
Get a box stock rifle, put a decent trigger in it, put on a decent set of glass and save the rest of your money for ammo. The only way you will get good is with practice. Then you will start to decide what upgrades you want on your rifle.
I have seen newbies to the distance game go and spend big bucks to have a rifle accurized, re-barreled and such only to see them get frustrated because their new tricked out rifle isnt putting 10 shots through the same hole at 600 yards.

It aint the rifle.

If you want to start drag racing you arent going to go out and buy a Top Fuel car are you? No- you are going to start out with a slower car and perfect your technique.