I suspect that this became customary for many piyutim in order to facilitate the continuity of the recitation, as if the tzibbur and the baal tefilah are continually finishing the other’s sentence. Actually, the piyut of V’chol maaminim actually makes more sense the way we recite it.
Is anyone aware that this particular piyut is far older than all the rest? It is reportedly written by Ezra Hasofer and Zerubavel. That dates it to the beginning of the second Bais Hamikdosh. Even the rest of the tefila dates back to the earliest known – Anshei Knesses Hagedola, who were towards the end of the second Bais Hamikdosh. Interesting.