R’ Wahrman z”l suggested that we can use the chiddush of R’ Ya’akov Emden that we have been discussing (here and here) to answer this question as well. Since in the case of ner chanukah the brachos are part of the definition of the cheftza shel mitzvah, they must go hand in hand with lighting -- a ner lit without brachos is not a ner chanukah. In the case of other mitzvos and minhagim, the brachos are separate from the ma’aseh mitzvah itself.
The GR”A comments on this din in S.A. that the lighting of menorah is like hallel on leil Pesach, which is said even according to the Beis Yosef in shul with a bracha for the sake of pirsumei nisa (the Rama disagrees and says hallel is only said at home as part of the haggadah). Apparently the kiyum of pirsumei nisa, even if done only as part of a minhag, warrants its own brachos.
I once heard from a talmid chacham that the reason we do hadlakas neiros between mincha and ma’ariv in shul rather than wait until after ma'ariv is not simply because if we wait until after ma’ariv everyone will run out (see M.B.), but rather because we need a “shem tzibur” to accomplish pirsumei nisa. Since ma’ariv is yet to be davened, that collective unit of a tzibur still exists at the time of hadlakah. However, once ma’ariv is finished and people are prepared to leave, there is no halachic “glue” so to speak that defines the group as a tzibur anymore. There may be 10 people standing around, but that does not mean they form a cohesive unit called a tzibur.