It is different to the UK. Care is obstetrician-led rather than midwife-led. You will see your doctor on each visit and he will provide all care and attend the birth. The system is more like that in America than in the Uk, (I've had babies in all three countries). Their midwives are similar (I would say) to American Obstetric nurses and not as highly trained as our midwives, hence the market for centres like the Mother and Child mentioned and for doulas, which provide things (services, but also childbirth culture) we are used to from the NHS.
State care, as someone said above, is very very good though you might find it culturally different (eg they are far more inclined to take the baby away and assume you need all the rest you can get. Just say if that is not what you want though and they will comply). The doctors mentioned above are well used to dealing with expats and tend to have trained and worked in the west so they understand us culturally. Those doctors who trained in Singapore will be excellent medical practitioners but their birth culture might differ - only a little though- You have insurance though so there is no reason to assume the doctors listed on it are not very good.
I don't know the reason for the limited list but if you ask you might find it is simply limited to those who practise at a cheaper hospital. If that is the case, you might be able to pay the doctor of your choice from insurance and have him/her deliver at your insurance company's choice of hospital, or you might decide to pay hospital fees yourself and deliver at your doctor's usual hospital.
Until you check with your insurance company the reasons for the list of preferred doctors and if there is any flexibility in it; and you check the options with your preferred doctors too, you cannot know what your choices are. I notice from the link I added TC Chang is already booked up for the Summer months so do ring. You might be pleasantly surprised and they are used to anxious mothers, new to their system. Remember that as it is a private system you get more of a voice and choice - it isn't quite such a set system as birth in our NHS.
If you run the names on your insurance list names past the midwives at Motherand Child or Parentline, it is likely they will know them. Singapore is a small island!
The cost of birth is not astronomical if you are on a comfortable expat package though there should be no need to pay as your insurance company is unlikely to use inadequate doctors.
I am sure your experience will be good.