Sounds to me like the "friend" has only read stories about hunting Alaskan/Yukon moose and the story where John Nosler's 300 H&H bullet failed on a moose.
Anyone who has hunted moose across Canada can tell you (as found in the 8 pages of replies here) that the 308 is perfectly adequate for moose, given proper bullet placement and within reasonable shooting distances.
Yes, they have been harvested with rifles from 22 caliber to 50 caliber (and most likely bigger), Does not mean that dome of those cartridges were great moose cartridges.
My own native grandfather harvested moose with the 22lr. He tickled 'em in the ribs, sat down, rolled a smoke, and then went and got his moose, who would have expired after a short run, and laid down as it wasn't feeling well. My native grandmother used slugs in her single shot 410 to harvest moose. Mostly because this is what they had on them at the time. My grandfather's main hunting rifle was a Marlin 30-30. He harvested more moose, deer, bear and beaver with it than I will ever shoot. I guess raising 13 kids on a trapline will do that!
To date, I have harvested moose with the 30-06, 6.5x55, 375 Win, 270 Wby, 338 Win Mag, 7mm Rem Mag, 270 Win, 7MM STW, 300 WSM, 358 Win and 6.5 Creedmoor, and archery equipment (recurve and compound bow).
I have to say, that rifle hunting moose with my leveraction 375 Win is the most fun! Furthest distance on moose with this rifle has been 120 yards.