GoKawiilThere are multiple reasons why Nintendo opted for its compact optical discs. The three main ones go like this: Games were harder to pirate, it meant Nintendo could avoid licensing fees, and it also led to a smaller machine that was cheaper to manufacture. Let's start with that last factor. While retro collectors have found GameCube games are so expensive in modern times, the console launched back in late 2001 for an absolute snip at $200. That made it seriously attractive for gamers on a budget compared to the PS2 and Xbox, both of which launched at $300.
Nintendo's choice of using miniDVDs was also an anti-piracy decision. Back in the early noughties there were basically no standard PC DVD burners that could read the company's proprietary disc tech. Though modders would eventually crack GameCube emulation with the Dolphin emulator, it was much harder to pirate GameCube titles than PS2 or Xbox games in the 2000s. That Nintendo's little console couldn't play DVD movies also meant the Japanese firm didn't have to pay the DVD Forum licensing royalties, which would have cost $20 per console. That's a whole lot of cabbage saved.
Those bespoke miniDVDs meant benefits for gamers, too. In keeping with Nintendo's tradition of committing to fast loading times, the GameCube's tiny discs could be loaded more quickly than PS2 DVDs, because their smaller size meant the console's laser had less distance to travel when reading them. Their form factor also directly impacted the GameCube's design. Due to their reduced size, it meant Nintendo could choose a smaller optical drive, which led to one of the most compact home consoles there's ever been.
Just look at the Xbox comparison. Sure, Microsoft's machine was a little more powerful than the GameCube. Then again, the OG Xbox weighed 8.8 pounds, while Nintendo's system tipped the scales at a mere 5.7 pounds. The GameCube was so lightweight, it could easily be carried around to a friend's house (hence the handle).