Congratulations on a real fun day at the range! Lucky dog!
The bigger the caliber the louder the shot, the longer it is to recover for follow up, the bigger the muzzle flash (flash supressors often hurt more than help by kicking up dust, disturbing brush hiding the shooter and hurting accuracy). .308 recoil can be easily managed by about
any shooter. Ammo is cheap, common, and reload supplies are everywhere. Plus it uses a short action. A shooter is giving up a lot moving up to something like .338 Lapua IMO.
Especially on this continent. Game just isn't THAT big. Deer and black bear haven't figured out how to wear bullet proof vests
yet. Imagin carrying a few hundred rounds, & loaded magazines of...