No other public office is a personification of the State like the presidency is. How the Prime Minister is selected.

From the Constitution Commission Report:

[ NO OTHER PUBLIC OFFICE - NOT THAT OF THE PRIME MINISTER, the Chief Minister or the Speaker of Parliament - is intended to be a personification of the State or a symbol of the nation's unity in the way that the Presidency is. ] 

In short:

The President is the personification of the state which is multiracial.

The Prime Minister is not.

The Prime Minister has a different selection process.

A person only becomes a Prime Minister after passing through “many stringent tests of leadership”.

(a) He must be elected as an MP in the general elections.

(b) He will generally lead and be endorsed by the political party which wins a majority of Parliamentary seats.

(c) He must command the support of a majority of elected MPs. These MPs would have had the opportunity to scrutinise his abilities and values closely.

(d) He must be a minister.

(e) He must be chosen by his peers in cabinet to be their leader.

As Mr Lee Kuan Yew said, for the Prime Minister to succeed, his peers must choose him and support him.

His race does not matter. His abilities are tested through a multi-layered filtering process before he becomes Prime Minister.

The Prime Minister is elected to govern the country. He is not the personification of the state.