Oh boy.... you really want to do this?
If the straight ninja sword is not an authentic ninja sword,
It's not, and there is no historical basis for it.
then why do so many Ninjutsu practitioners use the ninjato?
They don't. The only groups that do are completely fraudulent Western groups who base their "martial arts" on their own fantasies and what they see in movies.
What ninja school doesn't promote use of a ninjato?
All of the legitimate ones. The closest is a specific form of (curved, but shorter) sword used in Togakure Ryu, but it's still not the same as the movie-style straight bladed "ninjato".
Where did the ninjato come from and how did it inbed its way into ninja movies, books, comics, martial arts, etc?
A prop designer for the Japanese movies came up with them, as they were easier to make, and were an easy way to identify "good guys" and "bad guys"... not too dissimilar to the Western ideal of a "black hat" and "white hat" cowboy.... not based in anything accurate, historical, or realistic, but a simple visual shorthand to inform the audience.
Why does Stephen Hayes use a ninjato?
His early work did feature the movie prop, as he found one in a store in Japan, and decided to use it... that was not based in anything he was taught, but was an assumption he made before he knew better... and, sadly, attempted to defend such an action before finally admitting that it was baseless.
Why do other alleged masters and authors also make use and mention of a ninjato?
Who? Name me one legitimate "master" who use such a tool?
Why do some of them even have books with entire chapters dedicated to the ninjato?
Not knowing who you're referring to, the books your'e referencing, or the chapters contained, and taking into account your early defence of your reading of Ashida Kim (Ranford Davis/Christopher Hunter), it's entirely possible that you're reading the works of completely illegitimate (uneducated, fraudulent) "teachers" whose only claim is their own delusion.
What is the significance of it?
There isn't any. It's a movie prop.
Why is ninjutsu so intertwined with kabuki stage shows?
You have that backwards. And it's not a huge part of kabuki, you know...
What makes the ninjato a fantasy sword?
Besides the fact that it's a completely fantasy based item, with no basis in reality, history, functional use, and more? Really?
And do you believe that a ninjato can still be used to cause injury or death despite its quality and design?
Dude, that has to be one of the most idiotic arguments you've made... the fact that you can injure with something means nothing in this regard... I mean, I could do you a hell of a lot of damage with a pink plastic lawn flamingo... does that mean that plastic flamingoes were designed as a lethal weapon with a history stretching back centuries?
I'm going to finish simply. You have no idea what you're talking about. I recommend strongly you recognise this, and stop making yourself look foolish.