There are many good and logical reasons why the 243 survived and the 6mm Rem did not.............the 6mm was based on the 7X57 case, where the 243 is based on the 7.62/308 Win case with availability of millions of basically free brass from military ranges and police ranges. Secondly the 243 runs exceptionally well through a short action where as the 6mm, although marketed in a short action really needs a long action to attain it's full capabilities, where it now competes with the 25-06. The original 244 Rem introduction was with a 1-12 twist barrel which would only stabilize up to 90 gn bullets and was aimed solely at the varminting market. Winchester on the other hand envisioned their cartridge as a light game cartridge as well and put it in a 1-10 twist barrel and offered 100 gn loadings making it much more attractive to the eastern groundhog/deer hunters as a one gun does all cartridge.
When one goes back to the 50s you will also find that Winchester had a much larger following among gun nuts and gun writers of the day. Remington not nearly as much, so in them days Winchester could do no wrong and very little Remington did was ever noticed or written about. But the single largest reason was the first one stated...........availability of cheap/free brass and almost any cartridge based off a current military case is going to flourish, just look at the line up today.........243, 260, 7-08, 308, 338 Fed and 358 Win, all moderately to roaringly successful..............all for the same reason. You will not find a single mainstream cartridge in NA based on the X57 case, several fringe cartridges hang on, but certainly could not be called immensely successful, 6mm, 257 Rob, 7X57, 8X57 and 9.3X57...........why....lack of an endless supply of quality brass, cheap!!!
Look at all the successful and current cartridges based on the 06 case, same reason.
Oh and by the way Remington did not initially market it as a metric designation, so that argument is actually moot. Remington marketed it as the 244 Remington for the first years of it's life and only changed it to the 6mm when they reintroduced it with a 1-9 twist about 9 years (I think) later, by which time it was too late and the whole cartridge, regardless of twist was pretty much dead.