Would you say that God is one? Or do you think there are there three separate beings, Father, Son and Spirit? What One Person does, the rest are also present and doing. Jesus Himself says that when He says He can only do what the Father is doing. Thus, Mary can be properly called "Theotokos" because she bore the ENTIRE GODHEAD. The Logos is not "part" of God. He is FULLY God without any lacking or missing of part of the Godhead. Thus, the Word is FULLY God - and Mary bore God.
I agree with your second paragraph.
Regards
"Thus, Mary can be properly called "Theotokos" because she bore the ENTIRE GODHEAD"
No, the scripture is clear that the Father sent the Son. The Father and the Holy Spirit did not become flesh. What you are proposing is a form of modalism, somewhat like the "Jesus Only" pentecostals.
1Jo 4:10 Herein is love, not that we loved God, but that he loved us, and sent his Son [to be] the propitiation for our sins.