That depends strongly on your family situation and your job. For a single individual working in the city or a family with all their extended family nearby you could easily get away without a car.
But not everybody is that lucky and plenty of professions will require you to be in places where public transportation is not an option. An interesting reversal of intentions is happening between the cities of Almere and Amsterdam. Almere and Amsterdam are connected very well through public transport and many people will take the train into Amsterdam each day from Almere. The Amsterdam public transport system is reasonably good and services most of the areas with corporate activity. The fact that Almere has a large and skilled professional population caused some clever business people to move their operation to Almere. Initially those companies were staffed with people living almost exclusively in Almere itself. But now people from Amsterdam have started to commute to Almere in the morning causing a substantial amount of traffic the other way because the Almere industrial areas are not as well serviced with public transport as they probably should have been.
The best combination for such transportation that I've found is to take a small folding bike along on the train, this gets you within a few km of where you need to go and then it is an easy bike ride. But this option is not open to everybody, and it almost closed for me too due to a self inflicted bike accident.
Amsterdam is just about ideal for cycling, Copenhagen is the other EU city that really gets this right and Helsinki gets a lot of credit from me for being almost there. But the majority of EU cities have a very long way to go to make the bicycle the best form of transportation and even in Amsterdam bicycling is anything but safe (mostly due to scooters and taxis).
Amsterdam could probably get away now with closing off the center for traffic completely, say from Munt to central station and east to the Cruqius mill and west to Haarlemmer poort. They would have to build some more underground parking though because that would lose the main new parking garage at Oosterdok.
It's an interesting read but really does not answer at all. Don't see how having all extended family nearby is a factor. Rentals? Getting picked up by the family too remote from a public transport hub? You don't need a car as much as you think you do.
