Why did third parties develop multiplats on the PS2 and Xbox but not the Gamecube?
They didn't, not at first. The first 2-3 years, the Gamecube got nearly every cross-platform game, and wasn't left out any more than the Xbox was. It was only in the latter half of the generation that games started releasing on just the PS2 and Xbox.
Why? Because the generation started out with some semblance of parity. The Gamecube outsold the Xbox at first, just a little. It didn't have too bad of a stigma about its demographics. After all, Capcom planned to release its Resident Evil games as Gamecube exclusives. After the exodus over cartridge prices with the N64, most of the industry was expecting somewhat of a comeback for Nintendo, and had all their games lined up to be part of it - or at least to hedge their bets.
But as the generation moved along, there were several issues. Nintendo "kiddified" their biggest teen-friendly franchise, Zelda, with Wind Waker. Their marketing for Mario Sunshine was also a bit overly Teletubby-ish. The other consoles could play DVDs, and the Gamecube couldn't. The other consoles advertised online capabilities (limited though they were in actuality), but Nintendo actively bashed online and pushed their silly Game Boy Advance Link as an alternative. Nintendo refused to publish, much less hype, the specs of the Gamecube - even though it was a quite impressively powerful and efficient machine, and public perception was that the PS2 was more powerful, and the Xbox even more so.
As the demographics shifted back towards that persistent "kiddie" stereotype, sales of titles aimed at the larger 20+ demographic sold less on the Gamecube and more on the Xbox, and 3rd parties slowly phased out the Gamecube from their lineup - and the cycle fed on itself, as the Gamecube's sales slowed, the stream of games thinned, etc.
People like to cite that Nintendo's 3rd party woes go all the way back to the N64, but they don't, at least not in an unbroken line. They had a chance with the Gamecube to right the ship and muzzle the negative stereotypes, but instead, they ended up reinforcing them, and creating more.