Look, if you seriously have $1,000 to spend, you'd be nuts not to buy the brand-new Savage or Remington. We'd frankly be negligent to counsel you otherwise.
If your budget is extremely limited, however, AND you absolutely NEED top-level accuracy, then you could go for the P-14 with a custom .300 Win. Mag barrel. If you set that rig up properly, it will shoot. My 17 with a medium contour E.R. Shaw barrel in .300 Winchester cost me less than $500, all in, and it will regularly shoot less than 1 m.o.a. And an E. R. Shaw barrel is nowhere in the league of McLennan, Gaillard, etc. Not even close. My 17 is a bit ugly and heavy, but it is a joy to shoot on the range. When I burn this barrel out if a few years, I'll probably put on a McLennan or a Krieger and build a serious "tactical" rifle out of it. I've used one of those Boyd's "Varminter" stocks in the past on a 98 Mauser and found it to be excellent and cheap. I might pick one of those Boyd's stocks up for this rifle in the future to complete the package.
So if you are serious about spending somewhere close to $1,000, then definitely buy the factory rifle. The Savage would be simply impossible to beat for the price, and it's ready to go with no gunsmithing or fiddling.