The 7x57 is a great cartridge. I think the 275 Rigby as it is know in many parts of the world in the improved version chambered in a bolt gun is an outstanding hunting platform.
There is some debate about whether the 275 is a variant of the 7X57 or merely a higher pressure, lower bullet weight loading. It is definitely the latter, utilising a 140gn Spitzer as its primary loading at a time when 173/175gn RN bullets were the norm and loaded for Rigby's Mauser '98 action based rifles to rather higher pressures than the original 1892 level for the previous generation Mauser (small-ring) rifles. I've read a few times it has a marginally different chamber, but 7X57 ammunition can be fired safely in Rigby's 275 rifles.
The 7X57 can be loaded to considerably higher pressures than its forebears in today's rifles with good quality brass making it like the 6.5X55 SE/SKAN in this respect. Whilst SAAMI lists its maximum pressure as 51,000 psi, the European CIP gives it an MAP of 3,900 bar / 56,565 psi using the Piezo transducer measurement method.
For hunting, wouldn't the 7x57 Mauser with the same case capacity loaded to 7-08 pressures shoot the same bullet weights the same velocities?
Loaded to identical pressures, the 7X57 would produce higher MVs than the 7-08 simply because it would involve a heavier powder charge = more energy sitting in the case. As the larger case / charge reduces overall efficiency however, the increase isn't that great. If you model 7-08 v 7X57 in QuickLOAD, their performance is very close even with heavier bullets like the 'traditional' Hornady 175gn RNSP at their respective modern max pressures. (But lower performance from the 7X57 in US factory loaded ammunition at the very low 51,000 psi SAAMI MAP - and that's if all US manufacturers load up to that value which I'd consider doubtful!)
Conversely, load the 154 Hornady SST with H4350 in both cartridges to the CIP 7X57 maximum of ~56,500 psi, the 7X57 is predicted to produce 2,727 fps against the 7-08's 2,678 in 24-inch barrels. In full-pressure loadings the benefits of the 7-08 to the hunter are 7X57 equivalent performance but from a shorter cartridge that fits what is now the standard action length. The handloader gets equivalent performance from a few grains less powder per cartridge loaded.
The benefit of using a larger case in a given calibre / loading is often in running lower pressures to obtain the desired MV or ME. This distinction is best seen in large calibre African dangerous game cartridges. The classic English big game cartridges of pre WW2 used what were on the face of it overly large cases, low fill-ratios with Cordite type double-based propellants that saw lots of air around the Cordite charge bundles and very modest pressures allied to what many modern shooters see as ridiculously tapered case designs. It was done deliberately to allow cartridges to run at low pressures and thereby cope with extreme temperatures whilst almost guaranteeing easy case extraction. Bruce Wieland in his books on dangerous game rifles and cartridges is very critical of some modern US designs that started with the 450 Winchester Magnum and its straight-wall case that utilises a much smaller volume combustion chamber, 100% plus charge fill-ratios, and much higher pressures than 'traditional' designs all to get a 450 that would fit the Model 70 rifle.