If you're looking on the budget side, I'd recommend the savage axis. I have one in .223 and it is pleasantly surprising. I would imagine that a .308 would be fine, obviously more recoil, but its a fairly sound rifle.
I own the Edge (same thing before they changed the name) in .308 and I'd have to agree/disagree with you.
The Axis/Edge rifles are actually great... for the price. That "for the price" part is what makes it what it is. It shoots straight, but it lacks a lot of things that a "good rifle" should have. The stock is complete garbage, the trigger is heavier than a buick, the action isn't smooth, and since it has such a thin profile barrel it gets too hot too quick which reduces accuracy a great deal. So, the qualities of that rifle is that it shoots straight and it's cheap.
If I were to do it again, I'd have saved up enough money to buy a Remington 700 or a Savage 10. Something that comes with a good trigger and a halfway decent stock to begin with while allowing myself the opportunity for future upgrades to turn it into something amazing. Although now, you have the Ruger Precision Rifle...