Irenaeus let his imagination make connections not quite kosher. Eve didn't give birth to the original Adam [think about it].
By one man it is written, sin entered the world. The woman herself was not primarily charged with the crime, being punished for her association with it, by the pain of childbirth itself, though she too, as wife of Adam, was expelled from the Garden.
Now seeing Mary as Daughter of Israel, as spoken of by the prophets...that is safely enough more fully kosher, from a Christian perspective [with apology for borrowing the word kosher, to any Jew of faith whom may come along].
According to Hebrew law & custom, humans do not become diety. But we do see, in Jesus, God becoming for a time, man. Yet regardless of specious objections to the contrary, this praying to those whom have lived previously as humans, and only as human, themselves being merely created beings from the very onset (such as Mary) is too much much like the polytheism Abram was instructed to leave, in the first place! Those people prayed to various departed long-lived, powerful and influential kings (along with some ladies who became promoted to the goddesses once passing on from this world) as city-State god-protectors whom could be appealed to in times of need, or as objects of adoration.
One can be forgiven for praying to Jesus, for He was diety before having a brief period of human Incarnation.
As you touched upon earlier, "god-bearer" is more accurately translated than capitalized "Mother of God". For although it is said that Mary be not a Goddess...the attention paid to her is so much like goddess worship it not only approaches that same, but is difficult to discern much difference, upon more than a few instances of expression towards her, in RC theology and practice.
The title "mother of Christ our Lord and Saviour" or "mother of the Incarnation" can be more precise, though it takes longer to say.
Even Christ did not instruct to pray to himself, but more precisely to pray to the Creator more directly, in his name. A subtle difference, but one worth remembering.
It is your imagination, or maybe just your knowledge, that is lacking here, because Irenaeus was comparing Mary’s obedience with Eve’s disobedience. As for Jesus, we disagree. He was not man for a time,but for all time. He was and is, Mary’s sons and she, his mother.
She became the Mother of God, in which work so many and such great good things are bestowed on her as pass man’s understanding. For on this there follows all honor, all blessedness, and her unique place in the whole of mankind, among which she has no equal, namely, that she had a child by the Father in heaven, and such a Child . . . Hence men have crowded all her glory into a single word, calling her the Mother of God . . . None can say of her nor announce to her greater things, even though he had as many tongues as the earth possesses flowers and blades of grass: the sky, stars; and the sea, grains of sand. It needs to be pondered in the heart what it means to be the Mother of God.
(Commentary on the Magnificat, 1521; in Luther’s Works, Pelikan et al, vol. 21, 326)