In the US system, the chief executive (president) is elected separately from the legislature (Congress). Also, the legislature has 2 separately elected chambers (Senate and House of Representatives.) As such, you could argue that these two branches of government have the express will of the people and do not need a separate neutral head for any of them.
In the Singapore system (as with the Westminster system) the people only elect the Parliament, and from parliament is drawn the cabinet (the executive.) The good thing about this is that there is less likelihood of the kind of stalemate you see in the US. The bad thing is that it may be less representative of the will of the people. As such, institutions like a neutral Speaker has been built in to ensure all parties are fairly heard in Parliament and equitably represented outside it.
Also, it is NOT a good thing to have the speaker appointed from your party - it means you lose one voice. For this reason, Singapore's constitution actually provides for the Speaker to be appointed from outside Parliament should the composition of Parliament be so close that no party wants to give up anyone.