Gittin 62b: A Woman Conveying Her Own Get, And Begging The Question

I was troubled by the following gemara, but haven't researched it deeply enough to see if there is a resolution offered by Rishonim/Acharonim, or if my question is really a question. So all this might really be amaratzus.

The following translation pulled from my Rif Yomi blog, on Gittin 62b:

פשיטא: איש הוי שליח להולכה שכן בעל מוליך גט לאשתו
אשה הויא שליח לקבלה שכן אשה מקבלת גיטה מיד בעלה
איש לקבלה ואשה להולכה מאי
ופשטינן דאיש הוי נמי שליח לקבלה שכן אב מקבל גט לבתו קטנה ואשה נמי הויא שליח להולכה שכן אשה עצמה מביאה גיטה
כדתנן האשה עצמה מביאה גיטה ובלבד שתהא צריכה למימר בפני נכתב ובפ"נ
ואוקימנא בהולכה

It is obvious that a man may be an agent for conveying, for a husband may convey a get to his wife; and that a woman may be an agent for receiving, for a woman may receive her get from the hand of her husband.
A man for receiving and a woman for conveying, what?
And we resolve that a man may also be an agent for receiving, for a father may indeed receive a get for his minor daughter. And a woman may also be an agent for conveying, for a woman may indeed herself bring her get, as we learn {in a Mishna}, "the woman herself may bring her get, except she needs to say 'before me it was written and before me it was signed."
And we establish this as conveying.

I would read the basis of the question of the gemara as follows: A person cannot be a shliach for something that he himself cannot do. And so, how could a man be an agent to receive the get? After all, Biblically, the man does the divorcing, and he is not the recipient of a get given by his wife. If so, how can he act as an agent? And the answer is that there is a paradigm for a man receiving a get, so since he can receive a get Biblically, in general he can act as an agent for this.

There is an alternative, that this is not a question of whether a man ever is the actual recipient of the get, but rather whether he can act as an agent for receiving. And in the case of accepting the get for his minor daughter, he is acting as her stand-in. Thus, we have Biblical precedent, which we can extend to other cases.

But I would favor the former reading of the question, at least for now. And assuming the former reading, I have the following issue: The gemara asks whether women can act as shliach leholacha. And the answer is that a woman can, based on the Mishna which states that the wife may bring her own get.

But that seems to be begging the question -- proving something only by assuming as axiomatic the very thing you are trying to prove. After all, a woman bringing her own get would seem to be agency, rather than a din in and of the woman herself. As we see for example in Gittin 5a:

ותנן האשה עצמה מביאה גיטה ובלבד שצריכה לומר בפ"נ ובפ"נ שלא תחלוק בשליחות

Is there any basis for saying otherwise? If this is agency, where she is the shliach of the husband until she reaches her destination, then this is an instance of a woman working as a shliach. But we have not proven that can do it as a primary actor, for a get she is using to divorce someone else. So one can cite the Mishna, but what is the basis of that Mishna?

Perhaps we can say as above, that this is rather a matter of precedent, rather than one of whether the person is beTorat holacha of a get.

Another related issue which might also resolve it is what the definition of holacha and shliach leholacha is. Is this just traveling with a get under your control? If so, why should we need any kind of diyuk? This is a glorified postman, but one who must make sure that no one tampers with it. To coin an idiom (wink), the question should not be how we know a man, or a woman, can be a shliach leholacha. A monkey should be able to do it, or a parrot. Why not?

If, on the other hand, the shliach leholacha is also a shliach for giving the get -- such that one person or his agent, the shliach leholacha gives, and another person or her agent, the shliach lekabbalah receives -- then we are speaking of agency for giving a get. If I recall correctly, in the Jewish colony of Elephantine, women could initiate divorce. However, according to halacha, the man divorces the woman and not the other way around. If a woman cannot give a get, how could she be an agent to give a get?

I also wonder at the reason for needing to prove a man or a woman can be a shliach lekabbalah or a shliach leholacha/nesina in the first place. Why assume that they need to be betoras nesinas gittin or betoras kabbalas gittin in the first place? A woman's chatzer is koneh for her, and it is not betoras. Rather, it acquires property, and that property in this case is the get, which then has the effect of making the woman divorced. And we know from elsewhere that both men and women can acquire and transfer ownership. Let them be shluchim for this transfer of ownership, empowered by the husband or the wife, and once they are koneh or makneh, the divorce takes effect!

Perhaps we can combine all of this, and suggest that this was the question of the gemara. What is the nature of this shlichus? Do they need to be betoras nesinas/kabbalas gittin? Or even if not, such as a man for kabbalah and a woman for nesina? And the answer is given by showing precedent in all these ways, such that we know that they do not need to be betoras. And so we rely on precedent in these cases.

It may work, but I have reservations about this. Specifically, the focus on the man bringing his own, and the women bringing her own, seems to strongly suggest that the idea is agency only if you yourself can do it. Please let me know in the comment section what I am missing.