This question is essentially communal, and its ramifications for the community’s identity, cohesion, and avodat Hashem play a major part in any decision. Arriving at a balanced decision entails consultation with a local religious authority who is in position to judge what will be of most spiritual benefit to the specific community. In cases like these, communal customs and norms can have greater halachic weight than written sources. Our response is intended as a preliminary discussion of the general issues.
Singing an’im zemirot in synagogue every Shabbat is debated, as is whether the song should be responsive or in unison, and whether it is appropriate for a child to recite it. (See Teshuvot Ve-hanhagot 2:81.)
Why do boys so often recite it then?
A primary reason for having children recite it is as a form of chinuch (education), learning to be a chazzan for the synagogue (thus the tallit that many boys put on for this) and to connect with tefillot in a public way. Many girls would deeply enjoy the opportunity to connect to tefilla publicly as well, and there is no technical prohibition (with the exception of a minority view that a child’s voice raises issues of kol isha).
At the same time, reciting an’im zemirot would be chinuch for a prayer-leader role that a girl will not be able to play in classic Orthodox synagogues once she gets older, and over time might highlight that disparity in a way that outweighs benefits of short-term parity. Chinuch for a child that is at odds with adult observance is usually inadvisable. (This may be different from the question of what girls and boys do in prayer in co-ed preschools and primary schools, since the school setting is distinct from the synagogue setting.)
One possibility a synagogue might consider is creating an opportunity for a group of younger children, girls included, to lead a prayer together at the end of tefilla, like adon olam. (Or perhaps even to consider changing the minhag of how an’im zemirot is recited in the synagogue to accommodate a group of young children.) In this way, the girls get to be part of a group publicly sharing love of tefilla, in a manner that resembles how they will participate in tefilla be-tzibbur in the future. There are some boys who are reluctant to take on a solo role who may benefit from such an arrangement as well.