I don’t see where in that passage the woman does any “crushing”. She just gets help to run away, and help to be protected from the beast’s attacks.
But the verse in Revelation 12 refers to "the woman and her seed", just as the verse in Genesis says "the woman and her seed". This is the clear connection between the woman in Genesis and the woman in Revelation 12. The 'woman' in both books is preeminent and central to Christ and the church, and Satan wants their destruction. Both references are to the Blessed Virgin Mary. See in Rev. 12 at how desperate the devil is to strike the woman, (kill her, but fails). It's interesting to note that in Revelation the Devil "persecutes" the woman, this could be Satan 'bruising her heel', it is a serious attack but not fatal. So there are many correlations between "the woman and her seed" in Genesis and "the woman and her seed" in Revelation.
It says of the woman in Revelation 12:
"And she brought forth a man child, who was to rule all nations with an iron rod: and her son was taken up to God, and to his throne".
It further says:
"And the dragon was angry against the woman: and went to make war with the rest of her seed, who keep the commandments of God, and have the testimony of Jesus Christ".
There's no doubt that the woman in Revelation is Mary, and that "her seed" is Jesus Christ and the Church. Since both Genesis and Revelation refer to "the woman and her seed" being attacked by Satan, and in both verses the woman's wounds are not fatal, the only question is 'how can they not be the same woman', and how can that woman not be Mary?
You also raised the question of why Mary, and not God Himself, would do the 'crushing'. My answer is that God chose Mary to be the human instrument through which Jesus would come into the world when He could have done it Himself, without any aid from Mary, so why would God not use Mary to crush Satan's head? If God desired Mary's assistance to bring Salvation into the world, it follows that He would also use her aid to crush sin forever, to assist in ending sin forever. The N/T calls Jesus the "new Adam", and the Church teaches that Mary is the "new Eve". Eve's "yes" to Satan brought sin into the world, and Mary's "yes" to God's angel brought Salvation into the world.
I'm no biblical scholar, but if the Devil brought sin into the world through Eve, it follows that God would bring salvation into the world through a woman, and that woman is Mary. It says in Luke 2:35 that Jesus would be destined to "be the rise and fall of many", and that Mary's heart would be pierced by a sword also. This is a clear reference to Mary's cooperation with Jesus in salvation history. Perhaps Jesus allows the Virgin Mary to crush Satan's head because Satan pierced her heart as she stood helplessly watching Him suffer unspeakable torments on the Cross? No one knows the mind of God, but it has been Catholic teaching for millenia that Mary is the "woman" in Genesis.