I don't think we could say for sure that first party couldn't save the Wii U. The Wii U had a central selling point that consumers didn't understand or really seem to want. The Wii U's first party support was also much weaker, especially early on. Within its first year Switch had Nintendo Land, Wii Party U, Game and Wario, NSMBU, Pikmin 3, Wonderful 101, Wii Fit U, and Mario and Sonic. Super Mario 3D World just missed the cut off, but we could include it. That's really not a great lineup. Nintendoland and W101 were tragically underrated, but overall these games really weren't doing anything interesting.
Compare that to the Switch's first year lineup. Breath of the Wild, Xenoblade Chronicles 2, Mario Kart 8, ARMS, Splatoon 2, Mario + Rabbits, 1-2 Switch, Super Mario Odyssey, Pokken, and Fire Emblem Warriors. Even accounting for the fact that three of those games were available on the Wii U (which for BOTW isn't really relevant, because it was dead and buried by that point), that's just a much stronger lineup. Breath of the Wild on its own is probably better than the Wii U's first year first party lineup, and I don't think that's an exaggeration.
The Wii U didn't really start to get interesting stuff till the writing was on the wall. Sure Splatoon and Bayonetta 2 were cool, but at that point were you really going to invest in a system with a clearly bleak future? And it never got things like a mainline Animal Crossing, Fire Emblem, Kirby, Pokemon, or Metroid. And because Nintendo was no longer supporting a handheld line, you didn't have the same gaps in software that plagued Wii U, even when third party software is taken out of the equation.
The lack of third party software didn't help Wii U, but I think it was the least of its problems. The overall design of the system, marketing, and its mediocre first party lineup, particularly in the first couple of years, were the biggest problems. Obviously it's impossible to tell, but with Switch level third party support, I still don't see the Wii U hitting 25 million.
So, I don't think this really takes away from my point. It just shows the downside of Nintendo's strategy. It's more likely that one company is going to have a slump than all the companies in gaming having one at once. Which is probably why Nintendo's system sales seem to have more volatility than Sony or Microsoft.