Why should i hunt with a 300 min mag vs a 308 win or 257 wby for general big game(mule, elk, moose, cyotes)?

The .270 winchester is a fine big game cartridge. You could easily use it on deer, elk, or moose. It would be way too much gun for coyotes, unless you were trying to shoot them at 600 yards. For elk and moose, I like 150gr bullets like the Nosler partition out of a .270. The 130gr Barnes TSX would also be a good choice. Deer aren't that hard to kill and 130gr silvertips or core-lokts would be plenty.