The Danish and Norwegian pelt hunters killed and awful lot of polar bears with rifles in 6.5x55mm.
Every year in Sweden ~100,000 moose are shot and the majority of these are killed with rifles in 6.5x55mm. 139gr bullets are the minimum weight allowed by law for a class 1 rifle (Class 1 is required for moose and other large big-game) and 154gr is very common.
Swedish hunting system also has:
-hunter training which includes a practical shooting exam,
-hunter training that discourages low percentage shots such as quartering angles and long distances,
-a requirement that every hunter pass an annual marksmanship test (shooting at a standing and then a running moose target),
-a requirement that a trained tracking dog be available to track wounded animals within 2 hours of the shot.
-a very strong negative stigma that is attached to anyone wounding game
The Swedes tend to favour the 6.5 because the low recoil allows for accurate shot placement with quick follow-up shots and the meat damage is minimal.
The 9.3x62 is seeing a resurgence in interest, at least in the hunting magazines, as the number of wild boar harvested has hit >10,000 animals and the hype is around how tough they are. The vast majority though are shot with regular rifles in 6.5x55, 30-06, 308, or whatever.
There is some talk about people 'cheating' the annual proficiency tests by taking a 'six and a half' to the range and qualifying, but taking a 9.3 into the woods. "If they think they can shoot game with it they should be able to shoot it well at the range" is the reasoning, but reality says recoil keeps many people from shooting well, to the point that they distrust their ability to pass the exam with the bigger rifles.
If you learn to shoot straight and use good judgement your 6.5x55 is good for anything you'll come across.