My opinion is worth what you paid for it, so here it is. I have 5 Model 70's, 2 Custom 98 Mausers and a Montana Rifle Company 1999 all Weather ultralight stainless. Needless to say I am a fan of the 3 position wing safety and the controlled round feed with the claw extractor. The controlled round feed is quieter than the push feed, and functions in any position, even upside down. And the claw extractor just doesn't fail. Exactly why I favor them over all others.
Of the Model 70's, 3 are Extreme Weather stainless ( one purchased as an '06 immediately having the barrel removed to build a 280 Ackley - since Winchester doesn't offer it ) 1 Featherweight in 280 Rem ( it is push feed, but I won't part with it because it is so darn accurate for out of the box - light and fast if I'm hunting in brush) and 1 Super Grade in. .338 Win Mag from the custom shop in a rare "100 only" build before the CRF was re-introduced. The Mausers were custom builds, stocked on beautiful English Walnut.
The 2 stock Extreme Weather 70's were both very accurate out of the box, and required no adjusting. Tried a few different factory loads, and accurate to the point there is no need to reload for them. I personally love the new trigger. Crisp, light, and no creep. While it could be adjusted, there was no need, especially if you are hunting with it. I'm sure the '06 would have been just as accurate too, I just wanted a 280 Ackley on that action and stock. It is my primary hunting rig. If you chose that rifle in 7-08, I doubt you would be disappointed.
The Remington or the Winchester? You already know my choice. But is it the best choice for you, since caliber choice isn't an issue? The only way to be sure is to go pick them up, shoulder them, handle them. What feels best and comes to your shoulder/line of sight? I will add that many seem to have issues with the QC coming out of Remington these days. But you may get lucky and get a good one, if that is your choice. The same was said about Kimber, but many guys love them. They are a dream to carry, and I considered one in 280 Ackley myself, before doing the custom Extreme Weather.
All the hyperbole aside, if after handling both, you still can't choose...toss a coin! I'll even lend you one. Just be aware, my coin has two heads and they're both Winchester Model 70!