Please don't generalise. I am a Singaporean who is both very Chinese and very Singaporean in different contexts. There are Chinese Singaporeans who despise everything Chinese, but the trend of the times is moving against that kind of parochial and snobbish anglophile atttitude. And there are far more Chinese Singaporeans who feel Chinese than ABCs who feel Chinese.
I speak in a vaguely British accent (but not RP) because my father hates Singlish, and because it is good sense for a future academic to communicate well with all kinds of people. But I can switch to Singlish any time I want. Many better-educated Singaporeans who make the effort can do the same. It does not mean that I want to be a Brit.
The way I see it, when you are speaking or writing someone else's native language, it is a sign of respect to not speak it in a way that causes discomfort to him or her. That includes not mangling the pronunciation or the grammar. For the majority of Singaporeans who will hardly ever step out of the country, it's not an issue. But for Singaporeans like me it does matter. The same goes for the speaking of Mandarin/Putonghua - the standard at which most Singaporeans speak it just won't cut it internationally.
My comments are more for Heisui benefit rather than Yun, but quoting Yun as my following statements builds on his.
I would say it is only good sense for Singaporeans to be able to communicate effectively in English even if you are not an academic, Singapore cannot afford to be insular, a growing number of working adults in a White-collar job need to deal with foreigners at some point in the course of their work. Some, like myself, deal with them on a weekly basis, if not daily basis. English is the closest we have to a Lingua Franca and if you cant communicate your ideas effectively or make it easy for others to understand you, you are doing yourself and your company a grave disservice. We have a tendency to be closer to the aforementioned Received Pronunication when we try to speak proper English but that is only to be expected. i am somewhat amused if this is an indication that we are hoping to be a Brit especially since British implies nationality rather than an ethnicity. You may also be interested to know I use Mandarin when liaising with Taiwan or Mainland China.
As a Singaporean I can assure you the great majority of Chinese Singaporeans, even those whose command of Mandarin is vey poor arent ashamed of being Chinese. I am naturally referring to Chinese as an ethnicity here, not a nationality.