Well, I have to disagree.
I find none of those to be persuasive reasons to withhold lay leadership positions from women.
As for the priestly office, I’m not sure how being an advisor or electing a Pope are essential to the ordained priesthood. (Let’s remember that women already have SOME role in the Vatican. There are at least a few on committees — or whatever the precise terminology may be).
That’s a fair issue. Although you didn’t phrase it as a question, nevertheless it’s a question that deserves an answer.
The role of Cardinal goes beyond just being an elector for the next pope and (possibly) having certain administrative roles.
A Cardinal is also a direct representative of the Holy Father. Cardinals are entrusted with other functions which (to greater or lesser degree) require that they be capable of priestly ministry.
One example of this is that Cardinals have universal faculties to hear confessions. Obviously, one cannot hear confessions unless he is first a priest. That’s why, even though some rare Cardinals might not be bishops, they must be priests.
Cardinals are also sometimes appointed to preside at Beatification Masses. Again, he must be a priest.
Not all administrative functions can be open to lay people. When an office in the Church requires governance over those who exercise a certain function, the one who holds that supervisory office must be capable of actually exercising that function in the first place. In philosophy and law, we say that “one cannot give what he does not have.” or, as Fr. Z. likes to say “nemo dat quod non got.” For example, the Cardinal-vicar of Rome must be a bishop. Part of his role is to ordain priests for the Diocese of Rome. If he were not a bishop, he could never fulfill the duties of the office.
For the most part, the offices held by Cardinals,
even though they might seem at first glance to be mostly administrative, actually do require that the office-holder be capable of exercising episcopal (at least priestly) ministry. That’s often genuinely necessary—not just by coincidence and not just by the development of those offices over time.
Having said that, it’s still possible that the Church could allow for laypersons to be direct participants in the election of the next pope (history alone proves that beyond any doubt). Would such persons necessarily be called “Cardinals”? That’s a matter of semantics, not substance.