When I was a young boy and before I got my own 1st rifle I used my Dad's 300 Savage to kill several deer and 1 elk, as well as a few stock animals on the ranch. Now over 50 years later I have another Savage M99 in 300, and I am still killing game with it. I own a 303 British I made from an Irish Constabulary Lee action in the "speed" style. I am also killing game with that rifle now and then, and 20 years ago I had a 1916 Lithgow that was 100 original and looked unfired when I got it. I killed several deer with it too.
The reason I lump all these together is that I find the ballistics of the 300 Savage with 150 and 180 grain bullet and the 303 to be identical and so is the performance on game.
So I vote yes.
I think the 303 will do find for game up to about 700 pounds is used with with correct bullets.