Their claim is that each fragment gives you 30,000 CP: from my experience after collecting 120 fragments, only the four big game monsters (Ochu, Shaolong Gui, Immortal, and Yomi) and completing the game awarding 30,000 CP.
Additionally, paradox ending fragments each award 10,000 CP.
Most other fragments are usually the rewards for completing a side quest: the CP bonus you receive is just a quest reward and is somewhere between 200 and 1,000 CP. If you want to max out your crystarium grid (and your monsters' crystarium grids using monster materials), you'll be doing some grinding.
So while you technically can get 30,000 CP from a fragment, the site is indeed mistaken as there are only a small number of them that give that type of reward.
The site also claims each battle awards 1,000 CP off the bat: that's not even close to being correct. In first few areas, battles award something in the 25-100 CP range. The question about grinding has answers suggesting grind-worthy mobs cap out around 300–400 CP (which matches up with my experience), well below their claim of 1,000 CP being routine.