On the other hand, the quotes from the NT I had always taken as types and forsahadowings, not as reincarnation. "This is Elias" means "He is the man who was prefigured by Elias." "He is fulfilling the role predicted by Elias."
In much the same way, when a new pope is elected, the choir sings "Tu Es Petrus" -- "You Are Peter" --- but it does not mean the pope is a reincarnation of Peter, it means he is the successor of Peter.
I think the take away from the NT references is that while they knew Jesus was a local kid born and raised (as opposed to a mysterious stranger appearing out of nowhere), people debated about who he was. IOW, they debated over which prophet he - his neshama/soul - was. The gilgul of John, Elijah, Jeremiah, or some other?
Within that framework, John was the gilgul of Elijahu.