The Ryu Hayabusa situation is actually vastly similar to main protagonists from other major series. When closely examined, the fanbase of that character is basically the MC Purist type. Some other character examples of it:
Kazuma Kiryu (Yakuza) <--- this one especially
Dante (Devil May Cry) <--- this one especially
Goku (Dragon Ball)
Solid Snake (Metal Gear)
Geralt of Rivia (The Witcher)
Ryu Hayabusa (Ninja Gaiden/DOA)
All these characters I mentioned are essentially the blueprint of "don't change or fix what ain't broke", and they all have one thing in common: They have a generational fanbase since the days before social media where they are the sole main protagonist. No more no less. The demographic for them is typically huge since it insists on the absolute canonical and unchanging integrity of a fictional main character. Always coming up with a major plot revolving them after every sequel or so, basically nobody can take their spot kind of thing. Personally I'm mostly the middleman of this situation where as long as the writing is good for the plot, I generally wouldn't care to notice who stays and who leaves. I would usually only replace a main character is if their story is over, or the character itself is pretty bad. They can use the character 20 times in 20 new games if they wanted, but as long as the story is good to support said character that makes sense, then I generally wouldn't care.
In Ryu Hayabusa's case for the fans I can sympathize and understand. Though... I also believe that TN genuinely does not know what to do with Ryu Hayabusa at the moment, so instead of replacing him they keep him in the sides (which is the better option at the moment in my opinion). The problem with Ryu Hayabusa is that he is so incredibly powerful that you can't just randomly come up with a plot scenario of a high powered villain that threatens the world out of the box specifically tailored to him. And if there was, the outcome would be rather typical. He's just going to pull out the Dragon Sword, slot in the jewel on the sword like it's a phone sim card, kill whatever god it is and call it a day. It'll probably too straight forward so it only makes sense that they kept him in the sidelines to make room for new characters to experiment with on a understanding threat level. Least I'd like to believe so for this.
What makes you think Team NINJA doesn't know what to do with him if you don't mind me asking?
Now I'm not debating that those characters are important, although I would say the strong majority of them have counterparts that are just as good. I've always preferred Big Boss to Solid Snake, and I don't think people would think that's an insane belief. Vegeta IMO is ten times more important than Goku and sells just as much merch as that overrated buffoon. Let's remove that option because the writers and higher-ups care about him far more than the fans do.
Solid Snake &
Metal Gear Solid is now the best example when comparing Ryu Hayabusa with
NINJA Gaiden IMO. Sometimes it doesn't matter what the fan thinks.
Metal Gear Solid 2 was infinitely superior to
Metal Gear Solid 1 and anyone with a brain knows it or that has a knack for good taste knows it. The story, graphics and methos of it was all superior to the innovative, but subpar original and they did so by making Raiden the main character.
Did the fans cry and moan? Yes. Did it matter? No. MGS kept going without Solid Snake being the lead character. MGS2 is one of the highest rated games of all time, and definitely the highest rated game in the franchise history.
NINJA Gaiden 4's reviews matter A LOT. If the game reviews poorly there will definitely not be another one in our time of existence. It's got to have something since it doesn't have sales so, reverence, reviews and critical acclaim was a big thing that kept it around to begin with.