I am going out on a limb here, but I believe that everything God does has a purpose and this issue of men only priests has a reason deep in the human psyche. I know I won't explain it properly here because I don't fully understand it myself.
At the fall, Eve convinced Adam to sin. That implies Adam had an issue with taking responsibility - he just went along with what Eve had decided. Fast forward to Abraham; Abraham had a problem with simply "giving in" to Sarai and doing her bidding even though God made a promise to Abraham directly about his descendants.
My suspicion is that there is a deep "wound" from the fall and the result has affected men and women differently.
Men have a problem with taking responsibility by determining and carrying out their responsibilities aka God. In a sense, we use women as a crutch or excuse for not doing what we are supposed to do. We are taken in by their charms, often to our detriment.
Priesthood, for men, is a kind of service to be practiced autonomously, without interference from women to rebuild what was supposed to be the male psyche which is directly responsible to God, not responsible to women first. The weakness in the male psyche would be a weakness of character that fails to "man up" when called to by God; a weakness which then uses the influence of a woman to excuse itself.
Women, I suspect, have a slightly different set of character flaws resulting from the fall.
*Note: * The following could inflame some, but please let's keep this as a purely academic exercise. If you disagree, explain why you disagree. I am open to being enlightened and to discuss this further.
Women seem, in general, to have difficulty with accepting a subordinate role to anyone, including God. It has been the effect of the fall that women, in general, may have difficulty with accepting any matter that has been determined by someone else. Eve had a problem with this, so did Sarai. Women may not aggressively pursue the role of opposition to power, but do so in more passive ways until the issue is resolved favorably to them.
Perhaps the intent of God is to deny, intentionally, the role of "decision-maker" to women precisely because that is the role that is desired the most by them. Whereas, men have issues with taking responsibility for "caring" decisions and tend to leave those to women and excuse themselves by denying their responsibility for those kinds of "fatherly" decisions.
Note, I am not talking about actions that demonstrate male power in public ways. Men have no problem doing that, but my view is those kinds of public displays of machismo are ways of covering up a deep lack of authentic responsibility for others that men often shirk. Notice that Adam did not take responsibility for his decision to eat the fruit, nor did he attempt to dissuade Eve. He followed up by blaming her for making what should have been his responsibility to decide otherwise and, basically "threw her to the wolves" for making his decision for him, when he didn't have the courage to stand up for what was the right thing to do.
The position of priest, I suspect, plays on those deeply male flaws - flaws that tend to deflect the responsibility for "caring" onto women - in order to redeem all of male humanity by setting up a position that is only about responsible caring, the priesthood. Meanwhile, the denial of priesthood to women likely plays on the deeply female flaws that presume a kind of "right" to be right.
The question both men and women should honestly ask is, "Why would I want to be a priest?" Assessing those reasons honestly should be very enlightening. The second question, equally important, that should be discerned honestly, is, "Does God want me to be a priest?" If it is merely to have the power to carry out or inflict "my way" on others, even if that way is thought to be infallibly caring, that would not be a good reason to want to be a priest. A priest is, foremost, a servant to others. A conduit for the remission of sin.
To answer your question directly, my answer would be that a male only priesthood is intended to towards the undoing of the fall. Since the fall went from Eve (woman) to Adam (man), the redemption must, to be properly ordered in its undoing of the fall, move from Adam (priesthood) to Eve (Church). Men are to take up the responsibility shirked by Adam in order to carry out the undoing of the fall and bring the Church (bride of Christ) back to God.