I am thinking of going into business degree for university - either NTU or SMU, but everyone is dissuading me, saying it is too broad of a degree.

It really depends on why you're getting the degree what what you eat to do after.

Yes business is a very broad degree and some would argue that you can learn most of it by working a few years in various positions. If you don't want to do a more specialised degree then why not. You will still get the benifits if a good school and networking.

Since you mentioned part time, are you talking about taking a part time job or doing part time studies?