Singapore isn't a democracy in the same sense as the US or Britain. China is a not a democracy at all. So comparing the education levels of elected officials isn't really an apples-to-apples comparison.
In the US, a person's charisma, ability to relate to and attract the attention of voters, ability to build a political organization and ability to get things done within the political system define your success as a politician. Politicians start local, and build tight social networks as city councilmen, mayors, state legislators, etc. They then use those networks to get into Federal offices.
The great exception are candidates that skip the political process and reach high office by spending prestige capital (ie. President/General US Grant) or by spending lots of financial capital (ie. New York City Mayor Bloomberg). Even in these cases, these folks are actually buying access to someone else's political infrastructure.
None of this means that the US government does not have scientists working on policy. Scientists do work for the government as political appointees (ie. Secretary Chu of the US Department of Energy), as civil servants (ie. employees of NOAA), as contractors to government (ie. the RAND Corporation), or indirectly via unaffiliated think-tanks (ie. Federation of American Scientists).