My advice would be to try it for a month or so without and then decide..you will not have wasted much time if you then choose to get a car.
As you know a car is an expensive option here and taxis are plentiful and very reasonable...alhough if you do not live near a main route where it is easy to pick them up you will have to pay a couple of extra $$$ a time to call one to collect you. The school bus system here is very well organised and your children are old enough and (hopefully
) well behaved enough that safety is not such an issue (i.e. they don't need to be chained down in carseats or prevented running onto the main road while you try and get prams etc out of taxi boots)
However if you mostly catch taxis at busy times, they can be hard to get...you may be on the phone foreever trying to call one. Apart from peak work hours other busy times are when it rains and shift changeover (about 5pm I think?) when they all head back to Jurong or whenerever the base is and only take you on board if you are going their way. I wondered what was going on when this first happened to me as I couldn't understand a word they were saying as about 5 in a row drove off!
One other thing to note re cars is that parking is plentiful here - seems every mall or centre has something & although the costs can add up it is very convenient.