If many are struggling to define Singapore culture, don't worry about it. You're not alone.
I've had exactly the same discussions with the Kiwis who echo exactly the same sentiments as the sinkies... that they welcome immigrants provided they adopt Kiwi culture.
My retort is.. "what is Kiwi culture?"... "the Hakka and sticking your tongue out in a rude manner?". "No that's Maori culture." is the response.
"So you want the immigrants to drink lots of beer and puke and pee into the gutters every weekend?... or drive around screeching like morons and throwing beer bottles out of car windows?....or slur at each other at the barbie before keeling over and making a fool of yourself?
"Or is it sufficient just to drive a Holden instead of Merc or Beemer?". [Holdens aren't kiwi BTW they're OZ built].
At the end of the debate, there is still no defining trait that everyone can agree on. Some may say "love of Rugby" but only a minority of Kiwis are interested in the game in the first place and many prefer league which is an OZ thing.