The Orthodox Christian interpretation is that the essence of the Father and the Son is the same, but that they are still different persons. This is the Christian dogma of homousia that was affirmed at the First and Second Ecumenical Councils in the 4th century, and is stated in the Nicene Creed. At the First Council of Nicea in 325 the Church affirmed:
We believe in one Lord Jesus Christ, the Son of God ... that is of the
essence of the Father [ἐκ τῆς οὐσίας τοῦ Πατρός]
... being of one
substance with the Father [ὁμοούσιον τῷ Πατρί]
Theophylact of Ohrid (1055-1107), summarized the Orthodox understanding of the passage you cite as follows:
The Son is in the Father. This means that the Son is revealed in
the essence of the Father, and the Father is revealed in the essence
of the Son. Similarly, a king is visible in his portrait [if it is
well drawn], and the portrait is visible in his person. This is
because of the characteristics shared by the portrait and the king.
To show that His essence is the same as that of the Father, Christ
declares, The words that I speak unto you I speak not of Myself.
This means, "I speak only what the Father speaks. I have nothing of
My own that is separate from the Father. Everything We have, We
possess in common. We share one essence in two distinct hypotheses
[persons]. The words I speak and the works I do are the Father's.
Because Our works are divine, the Father is God and I am God: they
are the works of a single essence. Whatever I do, the Father does.
Whatever the Father does, I do.
Believe that I am in the Father, and the Father in Me
This means, "The very names Father and Son are sufficient to
indicate that Our essence is identical. However, if this does not
convince you that Our works are one in essence and honor, and that the
Father is manifested in My essence and I in His, then believe Me for
the very works' sake, for they are the works of God."
Explanation of the Gospel of John (tr. Chrysostom Press),
Perhaps, however, as you suggest some sort of argument supporting a different theology can be constructed on the basis of John 14:10 alone. This would be a very weak theology, though, would you agree? Any dogma proposed would have to find support throughout all of Scripture, without contradiction. We have for example, the testimony at the very beginning of John's Gospel (1:3):
In the beginning was the Word, and the Word was with God, and the Word
was God. He was in the beginning with God; all things were made
through him, and without him was not anything made that was made.
Furthermore, if the "Son" was only the "flesh", how do we interpret the tripartite division of body, soul and spirit? (1 Thessalonians 5:23). Did the "Son" refer only to the body, but the soul and spirit to God? How could the body only then pray to God the Father, as Jesus is shown to do in the Gospels.