Elijah will not return to Earth.
So further inference is your substantiation for use of the word "re-incarnation". You would be better off attempting to infer that because Elijah was the only prophet to be taken to Heaven in the manner of a fiery chariot/whirlwind something amazing happened to him. But there is Scripture to support that John the Baptist received the name "John" from his legal sire at the time via the angel Gabriel who was sent by God to Zacharias.
"The disciples asked him, 'Why then do the teachers of the law say that Elijah must come first?'
"Jesus replied, 'To be sure, Elijah comes and will restore all things.' "
The spirit never dies, but when Elijah bodily went to Heaven, that can be seen as the manner of his death. Our Blessed Mother was likewise assumed into Heaven at her death. But Christ had further work to do in the tombs. I have experienced supernatural blessing on the occurence of Holy Saturday, the only full day that Christ freed us from death. I have read of others receiving similar blessing on that most solemn of Sabbaths. Because it is God's plan that we live forever and not die, coming to know Him in the process, it is originally prophecied that God would send Elijah before the coming of the great and dreadful day of the Lord, lest God come and destroy the world.
As can be seen from your very quote, before Christ bore me witness that I had come fulfilling that prophecy, He plainly uses the future tense, "and will restore", knowing all things, inclusive to the Resurrection and Judgment. To the rest of us it is not given to know of the future as such, these things being sealed for our later benefit. When Christ was asked about His coming and the end of the age, He also plainly told His disciples by prophecying the signs and the further parable of the Judgment.
Even had I done nothing but run through the streets of Israel yelling "I'm Elijah, I'm Elijah!", I would have only been aggrandizing myself in a worst possible way. Through the grace of God, the Mercy of Eternal Salvation has been given unto us, and this in lieu of certain death.
Only Jesus Christ will return.
You're being quite the party pooper.
there's no prophecy that Elijah will return to earth.
Now you're binging. Christ admits to His coming, yet He always was, and will forever be. As we are made in His image, it is natural that a similar reflection be found in each one of us.