Picking up kids and doing errands with kids makes bicycling that much harder.

Nah, it's true. My walk to work is only possible because my oldest drops her sibling off at their school on the way to her school. If I had to make that loop it gets a lot more complicated, and even so it's the least complicated it's been since I had kids. There is a city bus to both schools, but the one that goes by the high school runs once an hour, the one to the college twice an hour but takes 2 hours, vs 15 minute drive. High school is an hour walk, not impossible and sometimes faster than waiting for the bus to arrive, but pretty bad with backpack and musical instrument, school bus system takes 1.5 hours.

The only reasonable answer here is car, until/unless the buses run on a reasonable frequency.