I don’t pretend to understand why God does anything. That being said, the best reason I’ve heard is that we are made in the Image and likeness of God. What we know about the nature of God is that God is One in Three Persons: Father, Son and Holy Spirit. If we are in fact made in the Image and likeness of God, then in some respect we are an Image of the Trinity - an icon, if you will, of the Trinity. Each person of the Trinity is truly God - consubstantial/or of one Essence, separate but not 3 Gods. The Father begets the Son and the Spirit proceeds from the Father and (through?) the Son.
Human beings are made male and female, where we are told the female was taken from a rib (or created after) the male - rather like begetting a Son. Childbirth where the same essence from the father and(through) the mother comes the child - much like the Spirit from the Father and Son. Somehow, to me at least, the Image of God in this way is crystal clear.
So, God made us, male and female. Both are human of equal dignity, but like the Father and Son, both being God, yet have different roles. That the Son and Father have different roles diminishes neither since both are fully God. That men and women have different roles diminishes neither as both are fully children of the Most High with dignity and inestimable worth.
It’s, to me, a simple lesson that we are each imaging God, but God Himself in His Trintarian Essence has Persons, who in the Unity of God, nonetheless have different roles. Men are not meant to lord it over women, just as the Father does not lord it over the Son - they love each other in their proper Person-hood. Though the Father is father and the Son was eternally begotton of the Father, both are God. So, a woman is not a priest…not because of any defect or inability, per se, but simply by reason of her role in the imaging of God. The Son is not diminished because He is not the Father.
As for choosing Peter and the rest of the Apostles, with all their faults, I say, “Take heart!” If Peter could be a great leader and saint, there’s hope for us. If we deny Christ, we can nonetheless love Him and be his Apostle and Disciple. An old priest I know once said, 'I think God chose Peter rather than John as Pope because He knew very few, if any, of us would have the courage to stay at the foot of the Cross when things got tough. Violent, fighting, lying, craven Peter - who ultimately came through it all in the end - is a role model most of us can match."