I started getting into practical long range shooting in 2009. At the time scope selection and action selection outside of customs was nowhere near what it is now. I went with a 700 5R, to which i eventually had bedded into an McM A5, then added an APA bolt knob and DBM with the intention to compete, though i'm not sure if PRS was even a thing then. It was still very much practical rifle matches of the likes of the Hide Cup and Blue Steel Ranch Safari matches, etc. There really wasn't a factory option to just buy into PRS and scopes were pretty much limited to Leupold, Nightforce, USO, and S&B.
I ended up putting most of my budget into a quality optic and later attempting to upgrade the rifle as i saw fit whilst shooting it. By the end of it i had sunk nearly 1400$ into a factory barreled action with no upgrades other than a bolt knob, not to mention the cost of the 5R itself being 900-1000$. I easily could've built a custom for what i've put into that gun. As most have noted Remington hasn't at all bothered to keep up with the evolving (rather exploding) precision rifle market. With the RPR and Tikka out there is no reason to buy a 700. By the time you upgrade it to have desirable features of most actions this day in age it ends up becoming pretty expensive.
308 i used to recommend as well, though other than barrel life in its favor, this day in age i'd say just skip it and go straight to 6.5CM. It's widely available at moderate prices and there is plenty of load data on it. Granted there is no substitute for practice and you will get more practice out of a 308 before barrel change. 308 is fine if it's more readily accessible and the subsequent barrel change associated with the 6.5 or 6mm cartridges doesn't bode well for the budget. Again an upside the RPR is that you can start with 308 as desired and later grab a barrel from LRI and change it yourself, rather than building another rifle.
I'd look at the RPR or Tikka T3, given your budget the RPR makes more sense to me. It's ready to go out of the box, other than putting an optic on it, you don't have to worry about DBM or an adjustable chassis/stock. If that EMT discount applies it's even more true.
In short if i was starting now, i'd have a RPR and the best optics i could squeeze at the perceivable time. Run the hell out of it.
My .02 cents.