What is it that the Swedes knew that make these guns such good shooters? The cartridge doesn't dominate the match results, 94s and 96s were nothing special in other calibers. Did the Swedes hand fit and assemble with only the best barrels and parts? I just don't get it.
6.5X55mm is an inherently accurate cartridge that has been used in Europe as a match cartridge for many decades in 300 metre competition. There was a considerable variety of improved sights available for the M1896s / M1938s including two different receiver mounted aperture models. It is still used today in the Scandinavian countries' inter-regional and inter-country matches using the Sauer STR 2000 rifle and issued ball ammunition.
It is the USA that has only recently woken up to the 6.5s' potential and by the time that happened, the 6.5X55 has been appreciated by relatively few people there thanks to a combination of its old age, low SAAMI pressure rating (45,000 CUP), and that it needs a long action.
The Swedish Mausers
are very well made and that's a help. They aren't the only military Mausers that can offer superb precision though if in good condition. There are many exceptionally accurate 7X57s out there too. (We Brits discovered that between 1898 and 1902 when we got our arses kicked in South Africa by Afrikaaner farmers turned Boer Kommandos using M1895 contract rifles, and your great great grandfathers found the same thing in Cuba around the same time in the Spanish-American war by regular Spanish infantry using the original 7mm Mauser, the M1893 Spanish rifle.)
I have a pair of Chilean long rifles, a DWM M1895 ('93 type small ring action as per the Spanish and Swedes) and Steyr M1912 ('98 action) both of which will put three shots into well under an inch and five into less than two inches off the bench. And since I can hardly see the crude sights and target together, that is really something. (Many years back, I read an American article somewhere - maybe Guns & Ammo, more likely one of the Gun Digest annuals - about the staggeringly small groups that a Brazilian M1909 7X57 produced off the bench when a small LER pistol type scope was mounted in place of the rearsight.)
I can't comment on the 7.65mm Mausers, but I've owned and shot many 7.92X57s including some unissued mint examples, the best of which was a superb Persian long rifle with a '98 action. None would shoot better than 2-MOA at 100, and more often 3-MOA or more in days when I could actually see through their sights. I don't believe this is a primarily cartridge design issue, rather the 7.92 chamber with its long tapering leade used to get the pressures down on very hot loaded cartridges.