Why would they want to though?
Spiritual power and privilege isn’t attributed to nuns (or any group of female religious in the Catholic Church) the way that it is to priests. Nuns (and all other female religious) aren’t even members of the Catholic hierarchy.
See, your problem is that you’re associating priests just with power and privilege. Priest are
not about power and privilege, any guy who becomes a priest for power is doing it for the wrong reason. What you’re doing is applying a secular attitude to religious life. Religious life is not about power.
Applying that same attitude, we mine as well say that Hillary Clinton totally failed at life because she
only became Secretary of State and never President. I would call her career a great success. What about career bureaucrat who become Under-Secretaries, are they worth less then the Secretary because they don’t have a place in the Line of Succession to the Presidency?
Now, I’m going to put on my mind-reader hat (and hopefully you’re not wearing your tin-foil one

) and guess you’re going to say “it’s not the same because they could have gone into that position”. That’s not the point I’m trying to make. What I’m saying is that positions of power is not what the Church is about.
When Cardinals go into a conclave, I’m betting the vast majority of them are praying “Dear God, please don’t pick me!”. Being a priest or a religious is about living a life of service, not about power.
Back to the issue of power, many theologians in this day and age are lay men and women. St. Catherine of Sienna and St. Teresa of Ávila are both Doctors of the Church, which is a recognition of great service to the Church (either theological or practical). St. Catherine
convinced the Pope to move back to Rome from France. St. Teresa reformed the Carmelite Order, which consists of both men and women. St. Francis originally wrote the Rule for St. Clair’s nuns (Poor Clair nuns), and she took the Rule and kicked the men out of it (Franciscans were originally to take care of the Poor Clairs I believe or something like that, and she took that out). Look me in the eye again and tell me now that women have no power.
I know a few women who have entered religious life, and I have the utmost respect for them. They are called to a certain type of life (contemplative prayer, serving the poor, etc), and that is why they enter religious life. Not for power and glory.
If you want power and glory, become a politician (so you can start by being a lawyer). Nobody who is truly out to serve God in the Church is out for power.
What could an average Catholic nun do that a Catholic male monk, priest, etc. couldn’t do?
A nun can do everything that a non-ordained monk can do (we often say that a monk is a male nun because people know nuns better than monks).
What this view is expressing though at it’s heart is that all people can be replaced by any other person; all people are inter-exchangeable. If I kill you today, we can take some other well-trained person in whatever field you’re in and put them in your job and call it a day. Your significant other can (assuming you have one) can move onto the next candidate and everything will be fine at home.
Obviously that’s ridicules. We can’t just replace 1 person for another because everyone is different. We have different life experiences and knowledges. You are uniquely you.
It’s not that they can’t do the work (preaching or whatever it is), it’s that
only men are called to share in the Priesthood of Jesus Christ. That’s how Christ implemented it starting with the Aaronic Priesthood in the Old Testament, with the Levites who served the priests, and with his chosen Apostles (the 12). Only men can be Persona Christi, in the person of Christ. The priesthood is intimately connected with the traits of Maleness; it’s why God is identified as Father. The priest gives the gifts (Sacraments) to the faithful, so they must be gift-givers: men.
Maleness is associated with the
initiation of the gift or the
giving of the gift. Where does that come from? Biology, specifically sex. In the sexual act, the man
gives the gift (the sperm) and the women
receives the gift. That is biological, there’s nothing we can do to change that. There are even arguments I’ve heard that a man who is a priest should ordinarily be able to perform sexually (can’t be impotent or sterile).
Please excuse the ranting, I got going a little bit there.