Yes, it is true that Singaporean children are not allowed to attend international schools unless they receive permission from the Ministry of Education. This is regularly granted for children who have one foreign parent or those who carry dual passports.
The international schools are expensive and beyond the reach of average Singaporeans, but I daresay there are many rich Singaporeans who make more than many expats. I think that many (though not all) Singaporeans prefer their own schools over the international ones anyway because they are seen as being academically rigorous and instilling a high level of discipline. It's a different mindset as to what education is about than what you are probably used to.
If you have a preschool age child, it is relatively easy to send them to a preschool with many Singaporean children.
For elementary age children, as a foreigner you may choose between an international school or a local one. This topic has been discussed at length, so best to search this site if you want more info on this topic. Check out the MOE website for how the enrollment process works. It is very different than the US--just because you live near a school doesn't mean your kid will automatically have the right to attend there.
If you have a child 12+, you may use a local school, an international school, or starting January 2005, local "international" schools affiated with Chinese High or the Anglo Chinese Schools. The local "international" schools are private schools that follow the Singaporean curriculum and fly the Singaporean flag. They charge less
than international school and accept a mixture of Singaporean and foreign students. (I think the mix is supposed to me 50/50, but I'm not sure.) I know very little about these schools, so best google to get more information. My guess is that most of the foreign students will come from other countries in SE Asia--Vietnam, Indonesia, etc.--as the Singaporean education system is highly regarded in the region.
I suspect the international schools would be more in line with your philosophy of education than any of the other options out there. Your children will be exposed to kids from many different cultures. Where it might differ from your experience in NY (or not) is that there will probably be less economic diversity. The kids don't qualify for free lunches at these schools!
Many of the international schools offer Chinese as a class either a few times a week or everyday. (Local schools offer it as one class daily with all other classes in English.) Your kids can get a pretty good exposure to Mandarin but are not likely to come away fully bilingual based on their school experiences alone.
Good luck!