It is true that a .308 is a good caliber to start off with since it's cheap to reload and factory match ammo is readily available offering the shooter a lot of practice for the buck.
ABSOLUTELY!
Before I bought my AWP, I looked at a bunch of calibers.. from 243, 260, up to 6.5-284 and 7RM. In fact, my original plan was to rebarrel in 260REM once I had my hands on the rifle. Now 7 months later, I'm 2600 rounds into the log book still shooting 308.
With 308, you get:
* Excellent barrel life compared to more overbore cartridges (any magnum, and any other 308 sized case with a smaller bullet).
* Ubiquitous match ammo, and many many known-good accuracy/match handload recipes.
* More rifles ready to go from the box
* Moderate recoil (compared to the magnums)
All of these mean you can spend more time learning to shoot, and more money on ammo or reloading components. You won't be rebarelling every 6 months or a year, you won't be fiddling around trying to find a good match load. When you run out of good ammo and have a match coming up, you can buy some Federal Gold Match, or Black Hills, and be good to go, whereas with some more specialized caliber, you might have to pull a couple late nights to get all the brass prepped and load a batch.
For practical targets, ie 6" - 12" at arbitrary distances, 308 is fine out to 700-800 yards if you're sorted out. 1000 yards requires more homework and judgement, but it's doable.
This isn't to say that other cartridges like 6XC, 6.5-284, 7RM, 300WM, 338LM, etc, don't have their place-- they do. Given the same amount of wind and ranging uncertainty, they basically extend 308 shooter's "effective range" by a couple to a few hundred yards. All those skills you develop shooting to 600 or 800 yards with 308 will transfer over to a more specialized caliber later.
Many long distance shooters keep a 308 rifle for practice, and a similarly-configured higher velocity & BC caliber rifle for match use. The 308 barrel lasts a lot longer and is cheaper to shoot, but he can switch to the other rifle to get its advantages when he needs them.