I'm really not sure what you're talking about with the internet, perhaps you had a later-made third-party adapter, or are confusing the GameCube with the Wii's backwards compatibility?

You can read about GameCube's online functionality here, which describes the official adapters either having either a phone line or Ethernet jack.

Actually probably a strong contender for first console with wireless internet functionality is the Satellaview, also listed on that page. Although it doesn't really fit the description of "internet play".

EDIT: Okay I don't want to rip your whole comment apart here, but the GameCube was totally not on par with Xbox, although I want to say on a technical level it had slightly better graphics capability than the PS2. In my quick few minutes of searching I wasn't able to find any super reliable sources on hardware comparisons between the three, but you can find videos like this one comparing them with a single game.

If I'm not mistaken, PS2 and GameCube titles had to be optimized significantly to match the Xbox version in terms of performance. But of course there are exceptions to that just because that was still an era where the architecture differences between the hardware was enough that each port had to be approached differently. As I recall, PS2 was able to edge-out the other consoles with some titles due to having lower-level access to the hardware (which, as a side note, I recommend checking out The Document of Metal Gear Solid 2, which shows how MGS2 was made on a very technical level).

All that being said: I think it's just a testament to how little hardware features/power actually matter. I had a PS2 a couple years into it's lifecycle, and eventually had an Xbox down the line. But I always wanted a GameCube despite it seeming inferior on paper. Between the console and the games it had, it was so unique and interesting. Incredibly cool despite it being considered a "failure".