Benny- the 7x57 will answer for most needs as you describe. There are better rounds for specific needs, as you related. The .375 and .366 calibres in several chamberings has alway been favourites of mine, but nowadays, unless guiding, I reach for the BRNO 7mm Mauser.
: I've tested and used 140gr. Combined Tech Silvertips for whitetail deer quite successfully and was impressed with the performance at high impact speeds. The 140gr. Nosler bonded cores didn't shoot well in my rifle, the the Silvertips thought they were match bullets. My load of 49gr. AA2700 might be a bit stiff for some actions, but was find in my modern rifle.
: For heavier game, the 160gr. Barnes Triple Shock with 51.5gr. H4831SC again, a heavy load beyond manuals, shot well for me, printing amny 1/2" groups with regularity. this bullet and laod wouod serve for all large game likely to be hunting with this ctg. Penetration is assured and I'd found the Triple Shock and XLC's which are baked with a blue coating, shoot better than the old style "X" bullets in my rifles.
: For standard bullets of heavy weight, the various 160 and 175gr. bullets like the 175 Spire Point, if driven at around 2,500fps or less, should serve admirably, and is an inexpensive bullet allowing much cheap practise.
: So you see, for most all game this side of the pond, and many in Africa, the old tried and true 7x57 still bears use. I would prefer my .375 or even the .458 for heavy game and tough hunting situations, but still reach for the BRNO when moose or deer are the target of choice.