6.5x57R and 7x65R are too small in caliber; bullet weight way too light for moose; sometimes too small for black bear too, if bear is wounded and charges you, for example. When hunting big heavy animals with such small calibers, you will get too many animals that you wound that get away. Yes, they will kill, if bullet placement is ideal, but in practical hunting applications, too often bullet placement isn't ideal, based on personal hunting experiences over 40+ years. Those who prefer needle bullets shot at high velocity for large heavy animals simply lack experience, in my opinion.
8x57JR will work well for whitetail, black bear, and moose, and the 9.3x74R will perform even better, especially for moose. However, in 9.3x74R it is best to have rifle around 7 to 8 pounds weight. 9.3x74R is a wonderful cartridge, probably one of about two or three best earlier day cartridges ever developed by the Germans. Use it, and you will understand why it is so well liked and popular in Europe, considered the Queen of cartridges. For moose, it is ideal. .30R Blazer ammo is too expensive, comparatively speaking.
My Dad was in Marine Corp. during WWII, fighting the Japanese, and from that point on, through his whole life, he loved the .30-06, telling me, when I was a young boy deer hunter in Montana, where I was born and raised, that one thing he loved about the .30-06 was that you could buy cartridges in any little two bit town anywhere in American; he was right. I have shot many deer, elk, and moose with it, often with 220 grain bullet weight for elk/moose, and was fairly satisfied with results (but, shooting similar animals with 9.3x74R, or .375 rimmed, I was even more satisfied with results). However, the .30-06 doesn't have a rim, so most people fear that it won't always eject/extract properly or well. In my collection of double rifles, both over/under and sxs, I have several chambered for the .30-06 and others in .270, most made prior to WWII; I have never once had a problem with any of them not ejecting or extracting correctly, firmly, so I think if the rifles were built well in the beginning; well cared for, one need not worry about such things.