Third parties are gaming icons(Bayonetta aside). Hell, the only 3rd party character we might ever get that is obscure is Geno. Bayonetta's not actually obscure either. But she's not a gaming icon either.
Nintendo characters are Nintendo icons and various NIntendo all-stars and even some obscure ones.
They do not correlate at all. Consider the fact that Sakurai has never mentioned one time that he chose a character due to their Nintendo history, but their gaming history or popularity instead. He was already good with Sonic getting in before he had a single Nintendo appearance. He just didn't have the development time for it. As soon as Brawl's work started, the next game, he immediately went after Sonic because of his major gaming status. Pac-Man was outright suggested by Miyamoto, but also was chosen due to his gaming icon status. Both of these are deliberate.
His literal response to choosing some Nintendo icon from FF over Cloud was "forget about console wars". It's evident he isn't looking for Nintendo-related 3rd parties. Those are coincidences and at best only one exception may exist(and we don't know the details behind Bayonetta's inclusion beyond her being worked on before the ballot got far in. She had to have had work done on her before it even started. That's key. That means there's more to it than the ballot alone. But until we have real information, there's no use in pretending the Nintendo ties were relevant). There has never been a single official case of where the Nintendo ties actually got a single character in specifically. Why are people pretending they matter when there's zero evidence to support that? A better way to put it is to get out of the Nintendo bubble and look at the bigger picture. 3rd parties are gaming icons first and foremost. If exceptions exist, it'll be because of the ballot or a reason we don't know. And there's no use in just assuming the reason is "Nintendo ties" when it's unfounded. We can't even cite a character who got in due to that, even if it's a likely reason for Bayonetta. But likely isn't proof. It's just a guess that cannot be backed up with evidence.
If it wasn't clear outright that Cloud was chosen entirely due to his gaming history and because FF is a juggernaut of a series(Sakurai never mentioned he wanted a FF character because it was on Nintendo, but because the series was important to gaming instead), Cloud is 100% based upon non-Nintendo appearances. He's intentionally the opposite of every Nintendo appearance. Meaning it clearly held zero importance. Especially since Sakurai never had a case where Cloud being on a Nintendo system mattered in choosing him. Let's remember that the situation didn't come up. He was asked about it, and all he gave was a "I dunno what I would've done", basically. Which is a very clear interpretation of his actual statement in context. Can't know if it never happened. But the reality is, he had a huge character that he was interested in having in because he was the biggest rpg protagonist of all time. The only actual thing he noted that was important, besides being an FF character(who he wanted in general due to the series having a ton of gaming history). The chances of him not wanting Cloud if he had zero Nintendo appearances is extremely slim. The worst that would've happened is SE saying no because Nintendo and SE never agreed on how to represent Cloud among any Nintendo appearances(basically, if the companies don't want to work together, that actually would've stopped Cloud from getting in. A real reason that holds any weight. Not a fan-rule).