As someone from sydney this is Sheer maddness. Sydney must be the most if not one of the most unsafe place on earth to ride a bike e or not, let alone one with unprotected small children. Sydney hates bike riders, the roads are hostile there are almost zero safe bike lanes.
Additionally you are exposing your kids to god knows how much vehicle exhaust fumes, right in their faces, sydney also has FA EV cars too.
Wow, Sydney sounds like a terrible place. Glad I live in a place that's safer and can get to work faster on a bike than a car. When I had to take my kid to elementary school it was about a mile away too (we walked though).