There are plenty of bolt action rifles nowadays for under $500.00, many of them already listed. The Savage Axis, the Ruger American, the Mossberg ATR, Marlin X7, Howa, T/C Venture, the market is full of them.
I bought two Savage Axis rifles this year for teenaged grandkids. The Savage is the only one I've found in a left-hand bolt, and wouldn't you know it, my eldest two grandsons are left-handed.
I'll probably buy another rifle to put under the tree next year, but that kid is right-handed, so the Ruger American might get the nod. I've had very good luck with both Savage and Ruger rifles and these two new rifles seem to have a lot going for them.
I agree with your caliber choice. The .308 Winchester is a wonderful all-purpose cartridge, fully capable of everything but the big bears. (and it may be capable of the big bears, but I suspect I'd want something bigger).