Before I offer my opinion, I'd mention that I'm not a mindless lemming just following the masses (over the cliff of blind ignorance and.....).
Also, for any hunting rifle/chambering the use of premium bullets, a load optimized for the barrel/rifle in a superbly accurate platform with appropriate optics are all presumed.
Like many here, I select chamberings/calibers based on anticipated range, what I'm hunting and ability to deliver more than enough power necessary to ensure immediate lethality.
A 20mm Anzio would not be entirely out of the question!
So, I break down hunting range into 2 categories - 0-700 yds and truly long range (my parameters). For me, truly long range is from 700yds out to my max., skill level in a rifle platform not too heavy to schlepp up the mountains.
For example, I'm planning 2 builds. I currently have a 7mm Rem mag - very similar to your 264 Win Mag in practical application. However, I'd like to get a little more ummmph for this "level" of rifle and will have a 7mm/300 Win mag built. I see this as the 0-700yds rifle for anything I'm going to hunt.
For the truly long range applications I'm planning a 338 Lapua "Improved" or some similar variant (there are a few to choose from - I like Kirby Allen's "338 Raptor"). I'm wanting this level of rifle to be able to easily achieve ~3,100fps with a 300gr bullet (Berger, SMK, etc).
I don't anticipate shooting past ~1,300yds so a 338 Lapua Improved variant will provide ample "hammer" to anything I hunt in a platform of reasonable weight.
The 300 magnum crowd will arrive shortly extolling the virtues of the .30 caliber of which there is some merit. However, for me and long-range hunting a fast .338 caliber bullet makes a convincing, well-rounded argument.
There are big, fast .375's available - something like the 375/408 CheyTac chambering, but for me they're not as "mountain portable" as I need. For me, the fast 338's have a nice balance optimizing power and portability.