I think I'll do some local trips on my bike too and see what do I get I guess.

Although I'd argue most of that isn't unique to families. Shorter trips are usually fastest by bicycle. (iirc up to 5km or so? Could be wrong tho) Having to change lines significantly increases travel time. Getting to an underground station with all the steps and elevators makes more sense for longer trips.

Although there is an important aspect of "sexist city design". Unfortunately I forgot the author who wrote about this (although I imagine at this point there are several). Basically public transit (as well as other elements of city design) are often built to mainly serve the "traditional man" or economic centres. Focus on getting people to work in their office in the business park in the morning and back in the afternoon, or to shop on the high street, or to the airport. But "traditional women" have to take many smaller trips that are less coordinated, like you show in the video. Daycare, paediatrician, groceries, taking care of an elderly parent the other side of town etc., at any random daytime.

A lot of that affects families, yes. But it also affects all of us if we want to do things other than work and shop. If you want to see your friends who live in another neighbourhood, just explore the whole town, do literally anything other than what has already been accounted for (because it can probably be profited of), then yeah, there might not be a convenient public transport link.

For example, all the well served lines in my city pretty much just go to the centre. When I searched for a new hobby I found a cool sounding club in a neighbourhood nearby. But I would've had to take transit all the way to the city centre, and then change to come back almost the same way. So instead of joining that club, I found a different club in the city centre. Not exactly what I wanted to do, but easier to get to. Which then strengthens the city centre even more, and makes the ties between the neighbourhoods even weaker...

I know it's not realistic to connect literally every street corner to every other point in the town directly, but I do believe that there could definitely be more interconnectedness between places. This would give all people more mobility options, and yes, help especially people whos responsibilities are more social and less corporate (like families). I'd wager it could also strengthen neighbourhoods economically and culturally, as people aren't forced to travel all the way to existing centres.
Walking and bikes will still always serve an important role for shorter distances, of course. Although for accessibility reasons getting around on just transit with minimal walk/wheel stretches in between should also be viable.

For the child related stuff: Where I live all schools have public bus stops, as most kids (6 years or older) come to school by themselves. A lot of kindergartens/daycare facilities are also near schools, so parents bringing younger children can use those buses too.