I've had a few precision rifles put together and gone through several barrels. You state that you want to build a precision rifle. For what purpose? Range use only, show off to your friends, compete with, varmint hunting, big game hunting, or what? The purpose will drive many of your choices.
I currently have a few factory actions that have been trued up and also have a some "custom" actions. I've used both types for hunting big game, varmints, and precision rifle competitions. If you want the best shooting, most reliable, I'd highly suggest you look into a "custom" action from Surgeon, Defiance, Stiller, Big Horn, or any number of other action manufacturers as opposed to having a Remington reworked. The immediate results may not be apparent but there is a reason the top competitors in precision rifle don't rely on a reworked Remington. I had to rely on one of my other rifles for the 2013 Precision Rifle Series finale which happened to be a Remington M700. The rifle shoots good but I had some problems that I hadn't had the entire year with my Surgeon action. Run your gear hard enough and you'll notice a difference.
This is just one guys opinion, so take it or leave it. Yes, the Remingtons can be just as accurate. There is more to a precision rifle than accuracy alone however.
Geb