A lot of those old mil surps took care of business.My dad killed a lot of deer with an old British.303 rifle.
OP if recoil may be an issue then some lighter reloads (like 125 grain bullet in .308 Winchester) might be the ticket - but may not cycle a semi auto.
I know in some rifle calibers, Hornady had "Custom Lite" ammunition and Remington had "Managed Recoil" but I haven't seen those on the shelf since before Covid.
https://www.hornady.com/ammunition/custom-lite#!/https://www.remington.com/rifle/managed-recoil/