pirsumei nisa through hadlakah b'tzibur

Lighting menorah in shul is done to fulfill the mitzvah of pirsumei nisa b'tzibur. The minhag in many shuls is to light between mincha and ma'ariv. Why not light before mincha, as the menorah in the mikdash was lit before the korban tamid shel bein ha'arbayim, or after ma'ariv, when it is dark? Rav Soloveitchik explained that the idea of tzibur means more than 10 people who happen to be standing around in the same place. There has to be some halachic "glue" that joins these 10 people into one entity and creates tziruf. Tefilah accomplishes that task. Rather than light before mincha when tefilah has not yet started, or after ma'ariv, when the tzibur is breaking apart, the lighting is done in between, when that shem tzibur is in place. (I have never been happy with this idea -- where do we find a makor that the kiyum of pirsumei nisa demands a formal shem tzibur? To the contrary, we find by other halachos of pirsumei nisa, e.g. kri'as hamegillah, that the idea of 10 is pattered after the mitzvah of kiddush hashem b'rabim, which simply demands a public audience. I am wondering if this idea of hadlakah b'rabim may in fact not be a chiyuv b'tzibur, but a chiyuv on the tzibur, and that may create the more formal need for tziruf.)