God is Spirit
One could argue that since God never changes God could not take on human form.
God is spirit, and those who worship him must worship in spirit and truth.”
(John 4:24 ESV)
Taken simply, this eliminates anyone who is human from being God. Yet interpreting God is Spirit, using our understanding of the physical and spiritual world leads to problems. For example, when a person speaks, it is their physical body which forms and causes words to be heard. Yet this certainty of the natural world cannot be used to define that which is spirit, since the Bible clearly states God speaks, despite being Spirit.
Moreover, the Bible has many events in which God is physically manifested. Man's first encounter with God is clearly one in which the experience is physical. If these are true, then God is Spirit cannot be understood to define God or narrowly to limit God's ability to interact with creation. Imposing a limitation would effectively mean God is not all powerful. In fact, if God does not change, then God's initial interaction with man must be seen as demonstrating God's intent for a relationship which includes physical interaction. We are in no position to use the Bible or our experiences, to define man's initial encounter as only "spiritual."
The "big picture" of the Bible describes how man's initial rejection of what God desires resulted in a temporary disruption, which God took upon Himself to correct. Where the man's initial rejection brought about a change determined by God, man's later rejection also brought about a change determined by God. Therefore the belief that God became flesh to restore what had been disrupted, is, in its most basic form, evidence of God's initial encounter and man's expectation of a final state.
"Only" True God
When Jesus prayed to His Father, and the Holy Spirit ensured that prayer would be made available to all who read the Bible, it is reasonable to assume He carefully chose His words:
And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent. (John 17:3)
αὕτη δέ ἐστιν ἡ αἰώνιος ζωή ἵνα γινώσκωσιν σὲ τὸν μόνον ἀληθινὸν θεὸν καὶ ὃν ἀπέστειλας Ἰησοῦν Χριστόν
The word translated as "only" is μόνος. This word is first encountered in the Greek translation of the Old Testament in Genesis:
Then the Lord God said, “It is not good that man is alone; let us make him a helper corresponding to him.” (LXX-Genesis 2:18 NETS)
καὶ εἶπεν κύριος ὁ θεός οὐ καλὸν εἶναι τὸν ἄνθρωπον μόνον ποιήσωμεν αὐτῷ βοηθὸν κατ᾽ αὐτόν
μόνος means "alone." When Jesus prayed He chose a word which also described the initial condition of the first man. Therefore, "only" does not mean there is no other. It describes the temporary condition which exists because God not only took on human form; He did so on the earth He created.
This verse is the clearest expression of the Trinity, and it comes from Jesus, not Paul or John. The letter John writes serves as further evidence John also understood this meaning of μόνος:
And we know that the Son of God has come and has given us understanding, so that we may know him who is true; and we are in him who is true, in his Son Jesus Christ. He is the true God and eternal life. (1 John 5:20)
οἴδαμεν δὲ ὅτι ὁ υἱὸς τοῦ θεοῦ ἥκει καὶ δέδωκεν ἡμῖν διάνοιαν ἵνα γινώσκωμεν τὸν ἀληθινόν καὶ ἐσμὲν ἐν τῷ ἀληθινῷ ἐν τῷ υἱῷ αὐτοῦ Ἰησοῦ Χριστῷ οὗτός ἐστιν ὁ ἀληθινὸς θεὸς καὶ ζωὴ αἰώνιος
When writing after Jesus' return to heaven and making a similar description of a belief leading to eternal life, there is no mention of μόνος. Why would John omit anything which is essential to obtaining eternal life? Because that which was separated when Jesus became flesh, is no longer in that condition.