The GameCube got most multiplatform games of its era and it was still a flop. Third party support's nice to have, but it doesn't necessarily mean that much.
But Nintendo is probably fucked without it. It isn't going to turn things around by itself but things will never turn around without it. As long as the third party support is still a problem Nintendo is shooting themselves in the foot every generation.
As for the Gamecube, to me it was death by a thousand cuts. If you took a bullet point list of the three consoles that gen the Cube finished last in most categories and was probably not number one in any of them (well I guess they were number one on wireless controllers). During a time where Nintendo had to show that the N64's problems were a fluke thing they showed up with a very lazy effort.
The Cube had:
- smaller discs
- no dvd support
- looked like a purple lunchbox during a time when Nintendo was routinely accused of being kiddy
- effectively no online gaming
- an irregular control layout that did not work for certain games, particularly fighting games
- memory cards at launch that were 1/16 the size of the PS2 cards but cost the same amount of money
- a "hits" discounted line that was $30 when everyone else had $20
Those last couple of bullet points are minor but then I think none of these things are a dealbreaker by themselves but combined it all looks pretty bad. And every one of these things was completely avoidable. It was like Nintendo was giving people a reason to write them off.
This is also the console that launched with beat-it-in-a-weekend Luigi's Mansion the same week the Xbox launched with Halo. And they lost Rare, which was their top asset on the N64, a year after the system launched. And they turned Zelda, which had undoubtedly the biggest amount of hype of all the titles pre-launch, into a cartoon during the peak of "Nintendo is kiddy". The timing of all of that was terrible. The Cube needed to be the "show me" console where Nintendo won everyone back and instead they found a way to at the very least do everything a little bit poorly. For me it the big reveal where I discovered how little self-awareness Nintendo has. I think they thought they could cut all these corners because they're FUCKIN' NINTENDO and everyone will just line up and buy their **** no matter what! They seemed very unaware of what sort of image the N64 had given them.