the most remembered games for these consoles usually include a good chunk of actual Nintendo titles. In the case of the Wii, they're almost the only titles worth getting that aren't shovelware.
Yes, I said that, but relying on third parties is not a good long term business strategy. Favoring the franchises you own and promoting those above all others while ALSO subsidizing your activities with third party support, but not making said third party support your bedrock, is a pretty sound business strategy if what you want is to survive in the long term off an independent market with little competition and not just make quick, hefty sums of money. SEGA failed because they were never as successful at this and and because their American branch simply thought more games and more developers who weren't SEGA was a good long term foundation for the company (spoiler alert: it wasn't).
Yes, but the Saturn was basically like the Nintendo 64 with most of its best games or exclusives being those developed by Sonic Team, AM2, Overworks, etc. who were part of the actual SEGA company. It had assistance from third party developers, especially in the area of popular RPG's, but it was not the third party machine that Genesis was in the United States. This however was enough for the Saturn to achieve success in Japan, but it was not enough for America for many reasons. One was the reluctance on the part of SEGA of America to promote Japanese RPG's and other weeaboo games which even though there was practically no third party Western support. The other was their reluctance or inability to promote actual SEGA franchises that weren't Sanic the Hedgeheg properly. The other was the abandonment of SEGA by third party, mostly Western game developers thanks to the fiasco with the 32x and earlier but less so with the SEGA CD and the early release of the Saturn which made many Western developers and retailers feel betrayed. But of course none of these harmed SEGA that much in Japan where the Saturn was quite popular. The loss of third party developers would never have been such a huge blow for SEGA if it could have still secured just enough of the North American market to make the success of the Saturn in Japan much more worthwhile. The Nintendo 64 was hardly in a better position than the Saturn in terms of a lack of third party support and yet was by no means a failure like the Saturn despite Nintendo driving away so much third party support. One reason, among several for this is that Nintendo had been since the NES & SNES era when it had plenty of 3rd party support, devoting considerable marketing and developmental resources to the franchises it actually owned.
The N64 lagged behind the PS1 and the Gamecube was also hurt by Sony's cheap DVD-rom drive in the PS2 just as the Dreamcast was. But Nintendo didn't suddenly go "oh, well let's pull out of making home consoles to focus on our strong suit: handhelds" because their home consoles under performed. Despite suffering many of the same problems as SEGA's own consoles, the Gamecube and the N64 still did considerably better and made just enough for Nintendo to feel that continuing in home consoles was worth it. Why? Because Nintendo developed itself as a brand with established franchises for one thing. Despite the problems Nintendo has faced with the lack of third party support, for those who enjoy Nintendo's many first and second party developed franchises, there is still plenty of motivation to have a Nintendo console, if no other other reason than to play these games. While third party support is important, it should never be the bedrock of your company because if you do something to piss the third party developers off and they abandon you, you'll end up like SEGA, who failed precisely because at least outside of Japan, they were never able to establish their own franchises on their own consoles as well as Nintendo was able to do worldwide, and thus they didn't have a net of consumers who were loyal to their brand or just big fans of particular franchises owned by them whose money payed for games & merchandise could help them at least pay their employees as they thought up a new strategy, which is the position Nintendo is in with the Wii U, which is still a better position than SEGA was in with the Dreamcast and Saturn, especially since the Wii U is also complimented by the 3DS.
tl;dr Nintendoom retards who don't know how to run a video game company think they know best what Nintendo should do