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Agree with everything you have said.
But God the Son is present in God the Father and God the Holy Spirit. God is a co-equal Trinity. So by definition the Mother of God the Son Incarnate is also God the Son Infinite and by definition the rest of the Trinity.
God the Father has no humanity. But He is manifested in the Incarnate Humanity of His Son

Jesus said ‘to see me is to see the Father’ in other words: the Father is manifested in the Son.As we say in that great Epiphany hymn: ‘God in man made manifest’
Incredible and awesome though it is, but the uncreated eternal unfathomable awesome omnipresent God paradoxically also has a creature He can call ‘Mother’!
That is one of the reasons why we pay such great honour to Mary, not just for the flesh she bore, as if that were not enough but also His relationship to the rest of the Holy Trinity: MARY THE MOTHER OF GOD
That also elevates Mary beyond the mother of mere Incarnate Flesh. Now that really is awesome

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