There is a principle recorded in the Mishna, Rosh haShana 3:8, to the effect that a person can only vicariously exempt others from their responsibilities if he shares the same responsibilities that they do. The mishna's language is as follows:
זה הכלל: כל שאינו מחֻיּב בדבר אינו מוציא את הרבים ידי חובתן
This is the principle: one who is not obligated in a matter may not exempt others from their obligations.
As a result of this principle, a deaf person cannot read the megillah (Megillah 2:4) and a woman, a slave and a minor may not recite Hallel on behalf of a man (Sukkah 3:10). In later halakhic literature, it is this principle that prevents a woman from leading certain parts of the tefillah¹.
¹ For examples, see Barry Freundel, "Kabbalat Shabbat: Recited by the Community; But is It Communal?". Pages 35-51 of Tradition 44:2 (Summer 2011), 45-46.
On the other hand, however, there is a principle recorded in the Mishna, Berakhot 5:5, to the effect that a person's emissary is an extension of himself (שלוחו של אדם כמותו). It is this principle that enables a man to have another give his wife a get on his behalf (Gittin 1:6, 6:7, etc), that enables a woman to be represented by another at her own wedding (Qiddushin 2:1), and that enables somebody to hire another person to separate his tithes for him (Terumot 4:4).
Who can be one's emissary? The Rambam (Hilkhot Ishut 3:17) has the following to say:
הכל כשרין לשליחות חוץ מחרש שוטה וקטן לפי שאינן בני דעת והגוי לפי שאינו בן ברית... אבל העבד אע"פ שהוא נעשה שליח לדבר שבממון הרי הוא פסול לשליחות הקידושין והגיטין לפי שאינו בתורת גיטין וקידושין
All are able to serve as representatives except the deaf-mute, the fool and the minor, since they have no sense, and the gentile, since he is not part of the covenant... but the slave, although he may serve as a representative in monetary matters, is exempt from serving as a representative in marriage and divorce since he is not within the scope of marriage and divorce legislation.
In other words, so long as one's representative is a full Jew (one for whom God's covenant with Israel and the relevant area of legislation both apply), it appears not to matter whether or not they share the same specific level of obligation. As such, it would seem that a woman may send a married woman to represent her at her wedding, a man may send a woman to deliver his get, and a landowner need not choose somebody who is him/herself liable to give tithes in order to separate his own on his behalf.
Given the above, why is it not possible for a man to say to a woman, "You are my representative; read Hallel on my behalf"? Unlike what was thought at the time of the deaf-mute, and contrary to the case of the minor, women are most certainly possessed of intelligence. If the only thing preventing men from having them serve as representatives in tefillah is the fact that they lack an equal maximum obligation, why does that not prevent us in other circumstances where we choose a representative?