I have also created a chat system using firebase and i was ran into the same problem you are in but i managed to solve that somehow i am sharing the code below please try that it should work:

      _scrollToEnd() {
if (scrollController.positions.isNotEmpty) {
  scrollController.animateTo(scrollController.position.maxScrollExtent,
      curve: Curves.easeOut, duration: Duration(milliseconds: 300));}}

Hope this helps please let me know if this does not help..