Traded for a like new Rem 700 Sendero in 25-06 a few years back. It has a varmint weight 26" barrel and shoots mildly and sub-moa. I like a more vertical grip than the oem HS Prec Sendero stock had, so restocked with an eBay HS Prec PST026 vert grip w/ adj LOP & cheekpiece. And I tweaked the trigger too.
It shoots just as well as a Ruger 77V I had of the same caliber, stolen decades ago, with which I could whittle away a car antenna stuck in the ground inch by inch.
Point being Rem makes one heck of an outa the box rifle, and Ruger did too.
Recently acquired a Ruger Gunsite Scout in 308. I really like it. First half box of rounds shot at a local indoor from a counter the wrong height in an awkward half squat resting on a bag and was very impressed for the 16.5" barrel.
I'll agree with others that 308 is probably the ideal caliber for what you want. What I have investigated like you re loving the quiet 300aac is that one can use very heavy projectiles (limited by barrel twist) with a pistol/shotgun powder; specifically 13grns of Red Dot for reduced load super-sonic and down to about 4-6 grns for quiet sub 1050fps loads on a par with the 300aac you tried shooting the same projectile.
As the Red Dot is so much bulkier than std rifle powder there's much less chance of double charging a case. There are other safety considerations; eg drilling the primer holes larger, etc.
Just got some 308 Lee collet dies and gonna try the Red Dot at 10grns and 6 grns with some std 150grn JSP's, 110grn JHP and some 100grn plinkers all of which I already had.
BUT, you can peruse it all yourself, see "The Load":
http://members.shaw.ca/cronhelm/TheLoad.html
PS; The Savage Axis it a great little bargain you could get for 300-ish in 223 or 308. Swappable barrels is something I'll eventually try myself for the possibilities of caliber and twist changes the easy headspacing and locknut system entail.
Those heavy bullets, albeit with rainbow trajectories, buck the wind and quietly still put some mighty impressive energy figures downrange.
The same might apply to CZ's sweet handy little M527 Carbine mini-Mauser in 223, or in 7.62x39 the Russian 30 cal, that might could utilize "The Load" for 300aac-like performance like the 308 can.
IF you go 223 and get a fast enough twist you can lob the heavier high ballistic coefficient 70-90 grain pills for long ranges.
