I'll have a try to explain ...
I live in the countryside, I have a network of small lanes on my doorstep and so my cycling is unimpeded by traffic as I rarely see any cars on the roads I choose to ride on, I try to avoid busy roads and A roads. Its all very easy.
When I drive my car I am often going to an urban area and as such these journeys are typically delayed by heavy traffic. It's only 7 miles for me to get into Canterbury but I usually have to allow 30 minutes because of the awful traffic in the town. It would be quicker to cycle.
The point I have been failing to get across is that when you drive you are likely to be delayed by an inconsiderate group of 20 cyclists on maybe 1/100 journeys, where as you are likely to be delayed by other cars on perhaps 95/100 journeys ... so cyclists are really not the problem ... the problem is their is too many cars for the infrastructure and there is no plan to address the situation.
So ... the media and politicians have conditioned the majority group (car drivers) to hate a vulnerable minority group (cyclists) for the problem (overloaded infrastructure) ... it's the same tactic that gave us Brexit and other unpleasant aspects in society ...
Cycling really should be part of our transport solution not a hate group, you give the example of the school run as an essential journey, that puts a load of cars on the road at commuting time creating more traffic. When we were all kids there was no school run we walked and cycled ... and guess what the classroom wasn't full of little fat kids either ...
Because cycling has been marginalised as a mode of transport and become a hate group it is little wonder mum does not want to send little Jonny off to school on his bike for the 2 mile trip ...
If you have been to some of the European cities which embrace cycling (I am sure you have) I am sure you would agree they are very pleasant places ... plus their health service is likely less burdened by obesity and mental health problems.
It's time we developed our transport system in the UK and with the advent of eBikes and eScooters there are much better solutions for clean and simple local transport than the car and you don't have to be a Bradley Wiggins wannabe to make a 3 mile commute on an eBike ...