You are missing nothing. You take annual holiday, or unpaid leave. If you have your partner living with you, you take it in turns and tend to have a negotiation about who's day can be rescheduled and who's can't.
If you use a childminder for before/after school care, they are more likely to be able to have your DCs on a day the school is closed until you can get home, but then you are reliant on only one person, so if the childminder is sick, you have no childcare, whereas the breakfast/after school club/nursery will have back up cover if one of the staff if sick.
There are emergancy nanny companies.
But, my oldest child is in year 4 now, there's been 1 day the school has been closed completely due to snow, 1 day the school finished early due to bad weather (and if you couldn't get to them earlier, they would look after your DC until the end of day), and 1 day when the nursery he was at before school needed to close early.
This isn't a big deal compared to covering the DCs being sick.
Nannies are better for that, as they will look after sick children.