The thing about the prophecy of R'hllor's religion is, as much as some people want to hang everything on those prophecies, the prophecies always seemed to be wrong.
Stannis was not the Prince Who was Promised. Neither was Jon, it turns out. The weapon of the Prince Who Was Promised that would kill the Night King didn't need to be bathed in the blood of a king.
Heck, even Melisandre's attempts to help in the Battle of Winterfell were useless. The Dothraki fire swords did nothing and neither did lighting the fire in the trenches.
About the only thing the God of Fire did was keep Dondarrion and Jon alive so that the Battle of Winterfell would be won.
Hate to say it but, all those in depth followers that based their idea on what would happen based on the God of Fire prophecies were swerved. Turns out the God of Fire was wrong about a ton of stuff.
Well, at least she correctly prophesized that Arya would kill the Night King, right?
I don't think that is the case at all. If Mel knew who was going to kill the Night King, she never would have backed Stannis or sacrificed those countless people including his daughter. She was doing that because she believe Stannis was the Prince Who Was Promised. The Prince Who Was Promised was the savior of mankind i.e. the one destined to kill the Night King. This is the show runners just taking some liberties and trying to come up with a logical reason why they altered the story away from Jon fulfilling his destiny. It just doesn't make sense. And as I said in my last post, if they knew Dany was going to turn heel then they should have kept the prophecy in tact or at least let Jon kill the Night King. The order will be different in the books, but at least if he had been the savior it is a lot easier to see why people loyal to Dany for years would just switch to Jon even before she went full on villain. The show runners have had very little for-sight in these things. They do things that look cool or for added drama, but those things undermine the story and the ultimate outcome.
Sometimes they do things for drama's sake, but I'm not sure this is a good example. Isn't it a theme of the books and the show alike that prophecies are difficult to interpret? A good example is the reveal by Missandei midway through that something everyone "knows" to be true just reflects a mistranslation - Azor Ahai is not necessarily the "prince" who was promised, and may actually refer to a princess. In GoT world as in ours, people believe they understand the true meaning of a prophecy but in fact they've misunderstood it. And who knows; the prophecies may be wrong or may not apply to the situation at hand.
To me, that's central to the show and the books. I continue to wonder why Jon has such a central role in the books; how does his story end? The fact it wasn't what people first expected, to me, fits right in.
And I think Melisandre's arc is consistent. She did believe that Stannis was the prince who was promised. Despite her magic, her age, and her gifts, we see that even she misunderstood what she was being told - or perhaps the Lord of Light was even playing a game with her. She was wrong; it's critical to the character and the story. Why should we be sure we know what the prophecy means or even if it's right?
As for Arya, my interpretation is not that Melisandre "foresaw" Arya killing the NK from the beginning, but that she got wisps and pieces that she misinterpreted at first but later put together correctly. She saw on first meeting Arya that Arya would close many eyes; much later, in Winterfell, when she sees Beric fall and realizes he's served his purpose, she puts the pieces together and thinks, maybe for the first time, that there may be a new possibility.