Can you shoot a factory rifle and win in a serious BR match? I don't think so, but I have been wrong often enough to hedge that bet.
Be advised that even among those with the very best equipment, there is only one winner anyway. Odds suggest that if winning is the only thing that can float your boat, you're going to have wet feet most of the time.
Consider goals with reasonable stages. A factory rifle will serve you well for several of those early stages. You can learn handloading fundamentals and advanced techniques before a factory rifle limits your overall performance. Shooting BR is different from..., just about any other discipline. Again, you can get pretty far along the main road before the factory rifle holds one back.
Here's my idea of a fairly good starter rifle for precision shooting. Try a
Savage 11VT from Dick's. They can be had in (at least) .223 and .308, my preference is .223 for the distances you have in mind. Yes, it's not a 6mm, but it is the chambering Sierra chooses to use for accuracy testing all their .224" bullets. It has a 1:9" rifling twist, which is adequate, but not excessive for the HDY 75gr HPBT Match (not A-Max) bullet. I load PPU brass with Varget and CCI BR-4's. I have shot several years worth of FV250 against other rifles like the Savage BR 6mm, custom 6.5x47, and loads of others. They beat me (of course they do...), but I continue to learn and improve. This rifle will hold you back eventually, but even then it will teach you how to stretch your limits. Only when you plateau will that ultimate rifle be appropriate, and you won't have worn it to shreds before you can really do it justice.
The Savage rifles are considered by many to be reasonably easy to replace a shot-out barrel on.
BTW, great optics can't be beat, but pretty good ones can get you far enough to hold off with the megabucks before you go for that ultimate rifle.
This is the scope I have on my 11VT. It's a highly adequate scope at an affordable price.
I own two of these scopes, and the other is on my custom Savage-based .260 Rem 1000yd F-Open rifle. My other mid-range rig is a Stag Model 6 Super Varminter AR, with a Weaver Classic V-24.
My rifle shoots handloads and the only mod beyond replacing the package rifle scope has been to lighten the adjustable trigger. BTW, I shoot off a bipod and rear bag. You can do better.
Greg