I note the 2 items on your wish list = #1 shoot out to 300m and #2 light weight. As others have mentioned 708 is very capable at ranges you state, & further. But you need the right projectiles & accuracy for 300+. Do you reload? Or rely on factory ammo ? Ideally in 708 I’d be picking 160 - 162 gn projectiles like ELDMS or X or 160 TMKs or you could run 150 ELDX, which are available in Hornady Superformance factory ammo. These would be a good choice if they shoot accurately from your rifle. Reloading will give you more options & the ability to tune loads to suit. If you don’t reload do you know some one that does ?
Chopping barrel to 16 or 18in will drop a bit of weight. An after market carbon fibre stock would be the best way to reduce weight & improve accuracy potential. What brand of rifle do you have at present? A Tikka in 708 and short bbl is a pretty light weight set up in the Tikka factory stock, and proven accurate shooters. I realise Tikkas aren’t everyone’s cup of tea, but it’s going to tick the box on your 2 wishes. If you want to go further down the light weight, longer range path then save your $ for CF stock. Just remember that shooting a light weight rifle accurately takes good skills & discipline on the part of shooter.
And there’s nothing wrong with a 270 either. But light weight 270, more recoil, is going to be more challenging to shoot accurately.
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