8mm Rem Mag (although poster then spoke of exactly why it shouldnt be)
Let me try again. I thought I explained why it wasn't
#1 338 Winmag had too big of a lead due to being around longer (Timing mistake)
#2 Remington Co didn't do enough to get it started right (marketing mistake)
#3 Takes a true magnum action (extremely minor rifle manufacturing complication)
None of those take away from the greatness of the cartridge.
Can't decide if you want the hitting power of the 338 Winmag vs the long range wonder of the 300 Winmag? Well with the 8mm Remington Mag you get BOTH. Not just 'split the difference' you get BOTH.
Even though the projectile is a little big bigger than the 300 Winmag, the case is bigger meaning more powder, meaning more power. Shooting bullets of similar sectional density even thought the 8mm versions are heavier the speeds are very similar giving you the same long range ability.
While not as wide as the 338, the 8mm is wider than the 30, combine with its slightly higher energy (again due to a larger case) it hits as hard as a 338 winmag.
Now, in the USA the 300 winmag, 338 winmag, and 8mm Remmag get SAAMI pressure caps of 6400, 6500, and 6500. A load that gives identical performance in a bigger case is doing it at lower pressures. In this case as you have a bigger case (full length improved 375 H&H case) at equal pressure you are doing more.
HOWEVER, the 8mm Rem Mag case is seen by most to be exceptionally strong and improved from it's parent in terms of shoulder angle, taper, etc in the best ways to the degree that the European version of SAAMI keeps the 300 winmag and 338 winmag at the same levels as their American counterparts but allows the 8mm Rem Mag to be produced at pressure levels of 6700 PSI. American Hand-loaders have uncovered the great strength of the 8mm Remington Magnum case as well and make some absolutely potent rounds, something that people reloading for the Winmag family can't do to the same extent due to needed to stay a bit lower in pressure.
Of course there is a bit of a 'chicken and egg' thing going on with the 8mm Remington Magnum. Most 8mm projectiles are designed with the 8mm Mauser in mind and can't always hold up to the 8mm Remmag velocities. Limited bullet selection turns many handloaders off which in turn keeps bullet makers from seeing it profitable to offer a wider 'fast 8' line-up.
Side Note. the 300 Winmag's case dimensions are less than ideal because right after the 338 winmag came out everyone wild-catted it to 300-338 winmag and then Norma formalized it with the 308 Norma Mag, so when Winchester came out with the same exact thing they had to slightly alter the dimensions just enough to be legally not the same exact thing...even though for all practical purposes it really was the same exact thing.
Side Note 2. Current SAAMI pressures for 300 winmag, 338 winmag and 8mm Remmag are 6400, 6500, and 6500. The 8mm Remmag comes from the full length 375 H&H case improved and necked down. The 375 H&H has a SAAMI pressure of 6200. It appears that in Remington's initial 8mm Remmag offerings they kept the pressures to the 6200 limit. Hence the cartridge initially didn't show it's true colors.