Snake was in because it was asked. It had jack-all to do with Nintendo. Sonic didn't get into Melee due to the fact it was too late into development. Nothing hints at Sakurai giving a rat's *** that Sonic had zero Nintendo appearances at that point. People need to let it go. Them being an iconic character was the reason they're getting in, not some Nintendo appearances. Bayonetta's your exception. If Master Chief isn't getting in, it has jack-all to do with some "hasn't had his game on Nintendo/barely was a costume". It's because Microsoft doesn't want their character on an opposing console. Which is fair, since Halo is 1st party and not meant to be on something beyond PC/Xbox.
No. They constantly promote different Pokemon starters over one another. It's a poor comparison and always will be because promotions don't work that way.
The "important starter" is chosen for your playthrough from the beginning. Gamefreak chooses the most important starter to promote. Decidueye barely was promoted. Primarina was barely promoted. Incineroar had tons of promotion. So did Greninja. That wasn't so much the fact they became the main starter for people to know, but Sakurai chose the characters who also were intended to be important in the anime. They were factors. He already is well aware those Pokemon aren't all treated as equally promoted. He actually pays attention to what Pokemon is promoted. There are exceptions, obviously.
The reason Pokemon isn't comparable, is that Sakurai also doesn't choose Zelda characters from concept art only. He actually set aside a slot for Pokemon and Pokemon only. He isn't going to do that for the Zelda series. That means he has no reason to choose a Champion that way either. None can stick out at this point, because all he sees is 4 equally promoted characters. Coincidentally they're all Stickers.
And you're wrong about Incineroar too. He had concept art that literally showed him hanging out with Ash. He already knew the character was going to be promoted. Only Incineroar was shown hanging out with other trainers. The other two were just shown by themselves(and pre-evolutions). It doesn't mean he would've guaranteed chosen Incineroar, but yes, he was well known as being a key promotional factor. Something the other two were not.
It doesn't mean it happens all the time with every Pokemon, but they've been doing it for a while now. And they're going to keep choosing a Pokemon to promote for the anime because it's an advertisement that works. People know the anime. Ash is way more known than Red too. It makes sense.
Greninja actually did not have art that promoted him as clearly being part of the anime. We don't even know if Sakurai knew that already. But he's an exception here. However, being as Sakurai doesn't look at concept art that early on when choosing characters except for Pokemon, it means, again, this is irrelevant to the Champions. Unless new promotional material pushes one as more important, they're just a group that belongs together and is hard promoted that way. It makes way more sense to just treat it that way and not try to make one more special when choosing Smash content. Having them all as Spirits is very logical.