I do think there are many people who do make "unnecessary" car journeys when they could walk. There are def a significant minority of people who just jump in the car to the shop/school purely becuase they are too lazy to walk or cycle. We can all make more of an effort not to use the car as much but for the majority of families it is not possible to do without one completely.
Public transport/walking/cycling is not always an option for 3 main reasons, disability, living in a rural area with no/little public transport or no public transport that goes to your place of work and it is too far to walk/cycle and get to work on time.
I have lived in several places and done several jobs where I have had to travel 30-45 miles to work (DH had to travel 60 miles in other direction) and this is not unusual. There was absolutely no way I could have got public transport to have got me there on time and I didn't even have kids to worry about. Yes we could have changed jobs (actually we both did and moved house so I didn't have to commute, although of course DH still has to commute but now its only 30 miles). However that is all about personal choice and TBH there is no way DH could do his job in the small town we live in (there are just no appropriate employers here) and he uses his car every day for work (it is a company car) as driving to various sites is part of his job.
Also, just because it is physically possible to walk doesn't mean that it is possible for people time wise. It may be physically possible to walk your dc's the 1.5 miles to school and then to get to work walk 2 miles in the other direction to catch a train for 45 mins and then hop on a bus for 15 mins followed by another 10 min walk to get to your office. However, it is not possible to do so in time to get to work. So you drive your kids to school, not because it is too far to walk but because you have to be at work at 9 and that means you have to drive.
In a world where we could all afford (or want) to be SAHM's, every village, hamlet and town had excellent public transport, there is no physical disabilty so we are all fit and we could live and work in the locality of our DH's employment (assuming their employment is not rural, on an out of town business park or involves travelling) then maybe we could do without our cars. However, until that utopia (or hell depending on your POV) is created, the vast majority of us we have to have at least one car.