It's funny, back in the days when it was announced that FF VII will not be released for N64 but instead exclusively go to PS I didn't saw it as a big deal. I knew that the FF-series is big in Japan and has some fans in the US but weren't even published in good old Europe. Just because of that fact I thought the FF-series can't be that big - otherwise, why would you miss the European market? Especially as it wasn't absolutely necessary to translate it - back in those days we were still used in Europe to have English in-game texts (but a translated description in the package - oh man how I miss those). I guess Nintendo saw it likewise and didn't see it as that big of a loss. They thought, FF VII would sell about as much as FF VI, with Sony's marketing power pushing it maybe all little more but no that much more. So, like myself, Nintendo didn't see a danger in losing FF VII.
As far as I know the mindset of old Nintendo, I think even after it was clear that FF VII is a massive hit, Nintendo just shrugged its shoulders and thought we can do our own big RPG IP or let someone else do it for us and we will have publishing rights. It think up to now, Nintendo think of themselves as the best gamer makers, period! No matter which genre - we can counter! That's really much their mindset. Nevertheless, I don't think that games like Mario Kart, Smash Bros, the divers Mario Sports games, Mario Party, etc. are meant as a counter to any other existing IP out there as Nintendo embraces the blue ocean strategy and are more focused on themselves than on what others do. Still, I think Nintendo wants to proof themselves and the gamers that they are the "masters of all genres". Splatoon (online-shooter) and Arms (innovative online-fighting game) are just their newest foray into new genres (they were not known for).
That's definitely not far from the truth, its a mindset of knowing they can do better which technically they did with Zelda OoT. FF7 was extremely important to the PS1 as Sony weren't known for making great games back than, they relied heavily on others making the games for them, basically doing an Epic Game Store and stealing the 3rd party games from under their feet of their competitors. While Nintendo just comes out and basically lets them take the 3rd party knowing they will do there own thing and as history has shown, Nintendo lives up to their own hype train on many occasions. With Rare by their side, they also conquered many other genres with Golden Eye/Perfect Dark dominating the FPS market and Killer Instinct taking it up there with the best of them just to name a couple games.
Nintendo to me has always been about the quality, similar to Blizzard back in their days as well, delivering the best games in their respective genres.