Long post ahead...sorry Fami, but not sorry.
Breath of the Wild is this century's Ocarina of Time. A paradigm shift, regardless if people loved it, or hated it.
Most important though of the century is extremely tricky. Wii Sports helped garner that age of casual gaming with the Blue Ocean strategy, but considering it ended up as a fad I think negates its lasting impact. It might've been a big deal at the time, but I don't see many folks today saying, "Hey remember Wii sports? Yeah, that was an awesome game! Let's play that together right now!" It may have been important for its time, but I don't think it holds up as well as say Mario 64, or Ocarina of Time in terms of importance, and those two games are from the previous century. No doubt it's influential, and probably helped redefine how Nintendo as a company operated, or at least thought of things; I just don't think its impact is as long lasting as others.
Nothing from the Gamecube era was more important IMO, and that is despite how much I loved that system. The closest title from that era would've been Zelda Wind Waker, and that's really only because it was so controversial prior to its launch, and kickstarted so many memes. Some younger Fami here might not quite fully understand just how big of a deal it was when Wind Waker was announced. For some, it felt like a gut punch, and others were extremely curious of Nintendo's decision here. In the end, we forgot about "Kiddy Link," and "Celda," or even how cartoony it all looked, and found it became one of the best Zelda titles ever made, and still holds true even today. But, still not the most important Nintendo game of this century.
Besides Wii Sports, we had Super Mario Galaxy, which was a bit of a departure from what we got with Mario Sunshine (A fully orchestrated soundtrack for starters for the first time in a Mario game), and just an overall joy to play. Mario Galaxy 2 (a game which unfortunately I've never played) took what was great about it, and added in the 2D Mario level style, which was a nice throwback. I've read that it's still one of the best Mario games ever made.
Besides BOTW on Wii U, the only other game on that system (which coincidently got a re-release on Switch) that has any sort of impact was Mario Kart 8, but I don't think in the way most think. Ultimately, it is still good ole Mario Kart, but with some great race tracks, great music, and overall just great gameplay. You can truly just pick it up, and play at any time. It certainly evolved the franchise more, plus also tying it into other franchises (namely F-Zero), and its influence caused other companies to mimic/copy it in the form of Sonic All Stars racing, and Crash Team Racing as the two best examples. I've heard these are truly great games with some great content, and overall a blast to play. Mario Kart 8 though, especially when it got re-released on Switch as Deluxe, just blew up in popularity, much further beyond that I ever imagined. Over 60 million sold on Switch, plus the over 8 million on WiiU?! That one game between the two systems has sold more copies than Nintendo sold systems for N64, Gamecube, and Wii U...COMBINED!!!
If you ask me, Mario Kart 8 Deluxe is more important of a title to Nintendo than Wii Sports is, and that is despite Wii Sports still the crown jewel of best selling Nintendo game of all time. The truth of the matter is though, which game in 20 years are people still going to remember, let alone play? Wii Sports might certainly in the picture in terms of reminiscing of Nintendo's successes and failures, but I don't think it'll be a game that many people will play decades from now. Perhaps it'll have some resurgence at some point in younger generations, but I believe more people will still be playing Mario Kart 8 two decades from now.
Realistically speaking, you could argue this point for most Nintendo games. Many of them, despite launching even 3-4 decades ago now, can still be played, and are fun to play even to this day. Or even if we do go back 20 years, one example of a game that has such a dedicated fanbase today is Super Smash Bros Melee, something I never would've expected myself. I owned Melee back in the day, but got bored. And yet, the fact it's still played both casually, AND competitively is honestly quite awesome. I just wish Nintendo understood how important their fanbase was to them as a company sometimes...but I digress.
That leads me back to Breath of the Wild, and its importance, let alone impact. It may not have the sales of Wii Sports, or even Mario Kart 8/Deluxe, but I think over time, it is a benchmark for how developers will continue to follow in Nintendo's footsteps much like what has happened with Mario Bros on NES, Mario 64 on N64, Ocarina of Time on N64, ALTTP on SNES. The mark that Nintendo leaves has a ripple effect throughout the entire industry, and BOTW is felt more so than any other game as of now in this century.
It is quite possible BOTW may not always be that game though. There are still ~75 years left in this century, and I doubt most of us will be around to see it, and truly be able to answer this question definitively. I'll be 112 if I make it to the 22nd Century, which I won't say never, but that's a lot of over time in age if you ask me. I just hope I can walk around, and wipe my own ass at that age, and not be a vegetable.
I agree with the rest though you got a few we missing like the Pokemons it definitely possible we would never seen a Switch without Pokemon to keep handhelds at forefront instead of other hand option
