Your logic is flawed too.
The reason why Nintendo became what it is, was because they were basically alone in the gaming market after the crash, they held a monopoly with the NES, but Sega dwindled that with the Genesis and a better third party relationship.
Playstation sells because Sony is friendly to third party developers, the PS1 was a easy to develop hardware and used the CD format. Saying that it sells just because of 1st party is basically the mentality from the PS1 and 2 where they had near monopoly due to their large lead over the rivals, thus like 5-7/10 games were exclusive, then the previous generation came and no one decided to make exclusives unless they had received cash since development costs rose.
Xbox sold because it followed the same steps of Sega and Sony, by having a great relationship with the third parties, alongside a powerful x86 hardware, not to mention its Live MP service. Third party games sold better on it than the Gamecube, it even received M rated multiplatform games that didn't appear on Nintendo's console, and they represented a significant share of sales by proportion.
Nintendo had a long history, but it's mostly their games library that pushes their sales nowadays.
And their quality is strong enough to keep them going at their own pace, despite having released the Wii U.
Playstation and MS are a different story however.
Yes, there is the history that you mentioned, which is the main reason why Playstation has such a strong brand recognition worldwide.
But Playstation and Xbox are too similar to truly be different from eachother in the way that Nintendo is.
So it basically boils down to two key-differences:
- Games library
- Features
This gen has clearly proven what the consumer actually wants: games.
And again, this is where Xbox is severely lacking.
Their biggest problem is that 90% of their games library is also available on Playstation, where vice versa it's more like 75-80%.
Then there is the gap in quality of the games and Sony has 1st party output on lock. That's not even up for debate.
Throw in the fact that MS decided to release their 1st part exclusives on PC too and moving towards streaming, what else is left for Xbox?
Btw, I put 1st party between brackets, because there are more exclusives than just 1st party.