look at a 300 case next to a 308 case. Other than a shorter neck, they're virtually identical. I'd be surprised if you could tell a difference between the recoil of a gun chambered in 308 vs an identical 300.
Interestingly, the 300 savage was the basis for the T-65 (308) case. The military started with a 300 savage case, and made some minor modifications to improve reliable function through machine guns. If you look into the history of the 308, you can find pictures of early prototypes. The earlier the prototype, the more it looks like a 300 savage