I think some of those are perceptual/cultural more than practical (that's not to suggest they're not real barriers, but perhaps not quite the same barriers people thought they were dealing with).
NL isn't that
different from here in that mums will tend to do more childcare than dads, yet actually more women cycle in NL than men. It's certainly more likely you'd have a sensible cycling route to school there, but plenty of British schools are quite bike accessible, but not bike accessed.
With the right bike you can take all the stuff (and children) required.
Showers are much talked about here because commute riding has a lot of crossover with sports riders, but again NL proves you don't have to work up a sweat. I used to deliver my 2 children to primary school on a 3-seater bike and then continue to work. I never had to take a shower or change when I arrived (and that's a 20 Kg bike up a fair sized hill, I just took my time in a low gear). And as with the cargo bike, electric options now available can be very helpful.
Skirts may not be as practical as trousers or shorts for riding, but it's certainly not impossible. The typical Dutch omafiets
has a skirt-guard as standard, but they're generally not necessary. My daughter often used a dress or skirt at primary (though trousers were an option she sometimes went with) and it was never an issue on the 3 seater, and once she'd graduated to riding herself it was still no problem:
You can also see in that they both have their bags with them, though the 3-seater had room for 4 so I could take their bags and still have plenty of space for any luggage I needed myself.
There will be numerous individual cases where a car is simply a better practical tool, but for many I suspect much of the problem is that with such a car-centric culture many people don't realise that a bike and everyday practical transport have quite the overlap they do. My wife or I rode the kids to school, and then escorted them (before leaving it all t them) not because we're eco-warrior bike activists, but because it was much less of a chore than driving and as a side-effect did everyone a bit of good. But this was easy for us, we already cycled by default so we knew it's not hard. The hardest thing is change, especially from a general default option. Why do so many people do the school run by car? Because everyone else does, that's what we do in the UK...
Often seen riding a bike around Dundee...