I would say keep the .303 because of the family history and because it shoots decently.
But I expect better than 2 or 3 moa out of a more modern bolt action. 1 moa or better is more to my liking. I would say that investing in a modern bolt action will be worth it. .308 is a good common caliber that can reach out a considerable distance. Its good for target shooting, taking medium to large game, and you can find ammo anywhere that sells ammo most likely.