I know there are the following 5.6X57 ,6X57 aka 6mm Remington, 6.3X57 aka 257 Roberts, 6.5X57, 7X57, 7.92X57 aka 8mm Mauser, 9X57, 9.3X57 and 9.5X57
They all predate 7.62 / 308 Win bar the 5.6X57 and improved 6X57 aka 244/6mm Rem, so used an existing continental European case and 'locally' well established / developed bullets. The UK and Switzerland aside, European deerhunters and competitors generally weren't keen on 30-calibre either until fairly recently.
in addition, the 30-06 and 308 were either banned completely in many European countries until recently, or if allowed required enhanced licencing checks, costs, and restrictions as they were classed as 'military cartridges' and not allowed for civilian use. (This legal factor was the reason behind many European 'wildcats', pistol as well as rifle, thereby producing the 9X21mm Italian originating design for instance as civilian ownership of standard 9X19mm Para pistols was illegal for many. The 9X21mm is identical to the standard 9mm Para bar the case being lengthened by all of 2mm!) The 5.6X57mm is a specialist continental European variation developed for the small Roe deer species where a 70lb animal is large, but provides the most common form of deerhunting across the entire continent. (Conversely, in the UK, Scotland allows 224 cartridges for Roe, but England and wales still insist on 6mm minimum calibre / >1,700 ft/lb ME.) So, there are many 'local' reasons for these variations on a theme.
As the nominal water capacity of the 7X57mm case is c. 59gn, only 3gn / 5.4% more than 308 Win, any performance enhancement is small even if run at 308 Win SAAMI pressures instead of the lower CIP 7X57 maximum. The AI version adds another 3 or 4gn capacity, so a possible 12 or 13% capacity increase. The rule of thumb is divide the case capacity change by 4 to get the % MV change, all other things equal (peak pressure, bullet, barrel length). So, if run at 308 max pressures (safe?) a 7.62X57AI would offer a 3-4% MV improvement, ie around 85-110 fps, on a 2,800 fps 308 loading. A fun project, but no great gain and possibly using a weaker case.
The Swiss 7.5X55 is a better bet as a beefed-up 308 and filler for the 308 to 30-06 gap. Although its CIP MAP is below that of the SAAMI 308, this is a proven strong case and has seen many developments and heavy loads in Swiss use where turnbolt rifles replace straight-pull or semi-auto rifles. (It's also an impressive long distance machinegun cartridge and remains in Swiss service use in the MG51 gun.)