Was John the Baptist really Elijah? If so, does that mean reincarnation is a biblically backed position? What do you make of these passages—
Matthew 17:11-13
11 Jesus replied, “To be sure, Elijah comes and will restore all things. 12 But I tell you, Elijah has already come, and they did not recognize him, but have done to him everything they wished. In the same way the Son of Man is going to suffer at their hands.” 13 Then the disciples understood that he was talking to them about John the Baptist.
Not Elijah in person, but John the Baptist, who was to come in the power and spirit of Elijah, Luke 1:17.
Between whom there was a great likeness in their temper and disposition; in their manner of clothing, and austere way of living; in their courage and integrity in reproving vice; and in their zeal and usefulness in the cause of God and true religion.
And in their famous piety and holiness of life; and in being both prophets; see Matthew 11:11 and that he is intended is clear from Matthew 17:10.
“Elias truly shall first come, and restore all things”: - He did not mean by this that Elijah was yet to come, for he tells them immediately Matthew 17:12 that he had come; but he meant to affirm that it was a true doctrine which the scribes taught, that Elijah would appear before the coming of the Messiah. To "restore" means to put into the former situation. See Matthew 12:13.
Hence, it means to heal, to correct and to put in proper order. Here it means that Elijah would put things in a proper state; he would be the instrument of reforming the people, or of restoring them, in some measure, to proper notions about the Messiah and preparing them for his coming.
Before the coming of John the Baptist the views of the Jewish leaders were erroneous, their expectations were worldly, and their conduct were exceedingly depraved. He corrected many of their notions about the Messiah (see Matthew 3), and he was the instrument of an extensive reformation, and thus restored them, in some degree, to correct views of their own system and of the Messiah, and to a preparation for his advent.