I don’t understand why a woman cannot blow shofar on behalf of a man, if he personally hears the shofar being blown? Answer:
Firstly, let us confirm that your assumption is not only correct but is unanimously agreed upon. Indeed the mishna (Rosh Hashana 29a) says that one who is not obligated in the mitzva of shofar is unable to blow on behalf of one who is obligated to fulfill their mitzva, and all agree that women are exempt from the mitzva of shofar because it is a time-linked mitzva (see Kiddushin 29a and Shulchan Aruch, Orach Chayim 589:3).
Your question is an excellent one, according to an understanding that many but not all authorities have. You seem to assume that the mitzva is to hear shofar blowing, not to blow the shofar. The Rambam writes in this vein in several places (see Shofar 1:1). The Tur (OC 585) provides two indications that the mitzva is to hear: 1) the language of the beracha is: "… He commanded us to hear the sound of the shofar." The halacha is that if one blew the shofar into a pit in a manner that he was not able to hear his own blowing, he does not fulfill the mitzva (see Rosh Hashana 28a). According to this approach, your question is logical: if one heard an authentic shofar sound, what difference does it make who produced it?
On the other hand, there are dissenters to this approach to different degrees. The Rosh (Rosh Hashana 4:10) cites Rabbeinu Tam as saying that the mitzva is to blow. He is not bothered by the language of the beracha, as Rabbeinu Tam’s version of the beracha was indeed "… on the blowing of the shofar." One can explain the need to hear the blowing by saying that while the essence of the mitzva is to blow the shofar, there is also a condition in the mitzva’s fulfillment that one hears it. Another simple question Rabbeinu Tam has to deal with is: if the mitzva is to blow, why don’t we all have to blow? One cannot answer that the ba’al tokeiah is our shaliach (agent) for the fulfillment of the mitzva because if agency was effective, then one could have someone blow for him without even being present to hear the blowing. In fact, the mitzva of shofar is a mitzva shebegufo, a mitzva that one must perform personally, just that we can apply the rule of shomeiah k’oneh (he who hears is like one who recites), which we find in contexts such as the reading of the Megilla. In other words, one who hears the sound of the shofar being blown is like one who produces the sound. Shomeiah k’oneh requires the reciter to be obligated in the mitzva, and thus according to Rabbeinu Tam we understand why a woman cannot blow shofar for a man. In contrast, the Rambam does not need to employ shomeiah k’oneh, as the hearing each person does himself suffices.
Yet your question does not disprove the Rambam’s approach. The mitzva can be to hear the sound, but not any shofer sound, but specifically a ‘mitzva- connected’ sound, which can be produced only by someone who is obligated in the mitzva. It is similar to tefillin. While the mitzva is to wear tefillin, not write them, only one who has the mitzva to wear tefillin can write them. What is very difficult for the Rambam is why the person blowing must have in mind that his blowing can be used by others in order for them fulfill the mitzva (Rosh Hashana 29a). After all, the sound is mitzva-connected because he is using it for his own mitzva or that of his friends, so why doesn’t everyone who hears that mitzva sound with his own correct intentions fulfill the mitzva by hearing it? (The Chazon Ish’s (OC 29:4) answer is beyond our scope).
According to a third approach, the essential mitzva is both blowing and hearing (see Minchat Chinuch #405). Therefore a man who hears from a woman is missing at least one element of the mitzva, being connected to the blowing of one who is obligated in shofar.