First the Prince Who was Promised and Azor Ahai are two different prophecies. The fact Melisandre (and now fans) blur that line underlines how murky and unreliable this thinking is to me. Second, I don't think the prophecy is where the show got its name. Beyond the obvious (White Walkers and dragons, the Wall and Essos being different extremes), to me it was always a story about Jon and Daenerys' rises, and Jon himself being born out of a secret union of ice and fire.
I have seen very little reason to believe in any prophecy in the books and show, because I always took a very skeptical view to that and so has Martin given his penchant for undermining prophecy left and right. Maybe Maggy the Frog was right, or maybe it was a self-fulfilling prophecy because Cersei turned Tyrion into an enemy by antagonizing him to the point of trying to kill hi multiple times and she antagonized Margaery to the point she invited the sparrows. There is a very non-religious reading of that.
The other main prophecy (two, in fact) were sold by a human-sacrificing witch who got it so wrong on Stannis that he (probably) ends up getting himself and his entire family killed. She already begins measuring Jon Snow as the next "chosen one," but there is a definite air of her making it up as she goes. I don't think prophecies mean anything in this world any more than they do in ours. It's probably why I didn't think about it at all when it came to defeating the White Walkers, because I never once believed Jon would have a flaming sword (much less make it flaming by running it through Daenerys like some of the more low-key-misogynist fans wanted) because it ain't that kind of story.
If you want foreshadowing, I do think there is plenty with Arya in retrospect. While I do roll my eyes a bit at Melisandre, you can just as easily turn to her "blue eyes" comment as her sizing up Jon after Stannis dies. There's also the fact she's trained from the start to say to the "God of Death," "not today." I actually think it lines up very well with her character, and I like it more because it isn't what we wanted. The thought of Jon Snow fighting the Night King just sounds like every other fantasy story.