I've killed many a deer with military arms -- primarily a '93 Mauser in 7X57 and an M1917 Enfield in .30-06. I've killed elk with Bigfoot Wallace, my custom '03 Springfield in .35 Brown-Whelen. I've also taken deer with a sporterized M96 Swedish Mauser in 6.5X55.
In general, military rifles tend to be a bit heavy and cumbersome, with less than optimum sights (with notable exceptions.) At one time, almost everyone I knew hunted with a sporterized military rifle of some kind. But the cost of sporterizing military rifles has gone way up, and the cost of those same rifles has escalated as well.
Nowadays, you can take a $600 Springfield, spend $500 sporterizing it, and wind up with a rifle that will sell for around $300 at a gun show. It's cheaper to buy a Remington, Winchester, Browning, Savage, etc.