Well - just to keep everything one off - sounds like you'll want a low power scope or RDO and the ability to send a follow-up shot quickly.
I just bought a second Beretta BXR-1 in .308 with the 16 inch barrel. Straight pull action - it's fast for follow up shots, and it's not hard to learn. I had the 30-06 version, and it was MOA or sub accurate with quality factory, easy recoil, threaded barrel, and decent fit for me. I had it as a back-up rifle for a Newfoundland bear hunt, but didn't need it, so tripped it for the .308 version.
The detachable magazines hold 5 rounds, so 6 in the gun if you top the magazine off. The safety will take a bit of getting used to - it's a three position push safety that locks the rifle closed, intermediate notch allows cycling, and the full forward is "off", but it lifts up against a spring tension going forward with your thumb for each position, then comes back to the locked position if you hold it up and relax. Not bad - just "different" than we're used to.
The .308 version has an adapter that will allow you to use the the Magpul LR/SR magazines in 10-20-25 versions. Bluntly - the 20 (or larger) rounder sticking out the bottom of the rifle is about ridiculous IMHO - but I could see the 10 rounder "maybe" being useful for pigs or predators. For the majority of my needs - two 5 round magazines will be more than plenty.