I believe the reason that the big bores are losing popularity is because of the tremendous development in bullet construction. A .30 will do more today than a .35 would do 40 years ago. (As far as stopping power and penetration)
As far as stopping power and penetration goes....no, not really. To illustrate why advances in bullet technology aren't much of a factor consider Bell. The 173gr 7mm FMJ round nose bullet that Bell used to take elephant with brain shots has better penetration than any expanding 30 caliber bullet today. Either you get a central nervous system hit and drop the animal, or you don't and they have to bleed out.
The 358 Norma was introduced in 1959, shortly after the 358 Win (and long after the 35 Whelen had been accepted as a swell hunting round), and the advantage of a bigger bore has always been the same, you get a more massive bullet which maintains more momentum upon contact with flesh. A high tech solid copper bullet in a 30 caliber will stay together great, but it won't produce any different effects than a Nosler Partition (introduced way more than 40 years ago) in terms of penetration or "stopping power"
So small bore rounds (8mm and under) are just not getting better, unless you count the high speed Magnums which can really take advantage of X bullets and Partitions. Of course those generally recoil harder than the 358 Win, so it's a bit of a wash (but you can't deny the popularity of the 300 Win Mag and 7mm Rem Mag). The word "magnum" definitely does sell firearms.
I can not think of anything I would shoot with a .358 Win. that I would not shoot with a .308 Win. For the record, if it is something that could eat me that I would be shooting, I would be using a much bigger case than the .308 Win. anyway.
The 358 Win falls into the same "medium bore" category as the 35 Whelen and 9.3x62, it's designed to take game cleanly with a minimum of fuss in areas where you might need to defend yourself from hostile man or beast. You don't need a 375 H&H to hunt Alaska blacktail deer, a 308 Win would do very nicely, but a 358 Win loaded with 225 to 250gr bullets would be a better choice for a "halitosis" encounter with a bear who decided he wanted your blacktail more than you did. Not that a 180gr bullet from a 308 wouldn't also save your life, just that anything you can do to stack the odds in your favor is probably a good thing.
But, if all you have is a 308, then all you have is a 308 and there is no point missing a hunt because a 308 isn't the best choice for defending yourself in bear country it is a dang sight better than harsh language and a sharp stick. Small bores can do it, as Bell and his 7mm took many elephants. But even Bell went to a 318 Westley Richards and remarked that many of his "inexplicable misses" went away. Sometimes bigger is better, but how much better is really hard to quantify.
Jimro