Why would anyone want to listen to Melisandre considering she got it so wrong with Stannis? She completely believed in him until she heard that half the men had abandoned them during the night. Thankfully Davos helped Robert's ******* son escape otherwise that would have been another useless sacrifice.
I think part of the problem is that her sudden escape from Stannis and what was left of his army implied she was a fraud.
I think it has a bit to do with the writing and acting. I don't think Van Houten is a particularly good actress and that was a tough complex scene to sell in a short amount of time. I think what the aim was, was to show that Melissandre realized she got it wrong. There was a "king" to support, but she thought it was Stannis and clearly it was not.
I think the storyline runs pretty simply from here.
If Jon Snow is half Stark/half Targaryen, which would be the practical premise as to why Ned lied to everyone about Jon's real lineage, including King Robert, then that means he's "fireproof" like Dany.
Someone will try to burn him since they burn all the dead to make sure they don't come back, and he will be "born" again as Jon Targaryen. Which fits into the old Maester's talk about let the "boy die, but let the man inside live"
Three Dragons, three riders. Dany, Jon and Tyrion.
One issue that needs to be resolved is to fight the White Walkers, any human army will need a craptastic load of Valyrian steel. Which means the entire arc of the mystery of that culture has to be answered.
Interesting finale, was almost a giant middle finger from the producers to GRRM to finish a book soon.