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historical note
in the mid 80s when LKY, then just over 60, was exploring his next step, the expectation at the time was he would move to the presidency – under the then constitution, parliament would appoint the president by a simple vote; he being who he was, the position would not be merely ceremonial but he probably felt not comfortable with a position that legally carried no power and was subject to cabinet/parliament decision; so a move was made to change the constitution and enhance the position; the public discussion led to his undertaking not to be the first elected president; Goh Chok Tong then closed the option of LKY being the second elected present, with a decision to invent the post of Senior Minister to keep LKY in the cabinet;
the subsequently elected president Ong Teng Chong found that the constitutionally prescribed new duties required certain information regarding government finances and asked for a complete set of data, which caused friction with cabinet and some civil servants. Further, the 2nd and 3rd presidential elections never took place, because there is a provision that a candidate must possess adequate administrative experience, the assessment being carried out by a committee of three senior civil servants; in effect, the 3-men committee “elected” President Nathan twice, by rejecting other candidates as not being sufficiently qualified.
if LKY had retired in 1990 and then started a newspaper column (modern idea would be blog) or a charity/research foundation, and he would have remained the most influential person in the country, taking into consideration that his son and his 2nd cousin (Tony Tan, currently the 4th President, elected in 2011 with with just slightly more votes than Tan Cheng Bok) were both in cabinet; if he had been a blogger posting articles daily, every important person in singapore, the cabinet ministers especially, would have eagerly read them as soon as they were posted.
Further, it would have been possible to make a different kind of constitutional change, by creating a senate with a proportionate electoral representation, e.g., if there are 20 members, a political party will send one senator for every 5% national vote it receives; this would have provided a good venue for retired ministers to stay around and have a voice on policy, without having to have power to pass or reject legislation, but merely being able to discuss and advise; by ensuring that small parties that cannot get any candidate elected in a first-past-the-post system, can nevertheless get some electoral representation, the system also makes the method of “unelected parliament members” unnecessary; undoubtedly, if LKY had stayed on as a senator instead of cabinet minister, he would have remained the strongest public voice in Singapore without exercising ministerial power nor receiving ministerial pay.
Yuenchungkwong's Blog 
note added on 8/5/2017 – a recent constitutional amendment imposes additional requirements on presidential candidates’ eligibility: the government may declare for each election the particular ethic group candidates must belong to; for the next presidential election candidates are required to be malay, so that tan cheng bock is not eligible; this might deal with the current situation, but in the long run I believe some solution along the line I suggest will be necessary
The PAP government has a big worry about the presidency: if in the next election Tan Cheng Bock runs one on one with incumbent Tony Tan, he is likely to win. In 2011 he lost only by a whisker, in a field of four candidates; since the other two were both opposition figures, if they had been absent their votes were likely to have gone to TCB who would have won handily. Many voters had the view that, with PAP as the majority party in parliament forming the executive of the nation, it would be nice to have a president with a history of different political affiliation to act as restraint. I dont think this attitude has changed
I always considered the elected president scheme to be a mistake arising from convoluted thinking; I outline the history at the bottom of the article. If PAP wants to ensure that the majority party also has its preferred candidate winning the presidency, then the present system need to be reformed.
The scheme I suggest requires a significant constitutional change: an upper chamber or senate whose presidency is ex officio also the head of state, with the same “second key” power of the current presidency. The senate shall be elected by proportionate representation, in parallel with the first-past-post system for the lower house. Small parties that cannot win any lower house seats can still gain some senate seats based on their total national vote; e.g., if the senate has 20 members, then a party winning 5% nationally would qualify for one seat. In this sense, the new system is more representative than the current one, providing for smaller parties having a voice. (Note: if the opposition total vote is 30% as in 2015, then 5% is not really a small number, and a party with this level of support deserves to have a voice in an elected forum) For the 2015 election, PAP would have qualified for 14 senate seats, which presumably would be filled by retiring ministers and lower house members, so that they can continue to fulfil an advisory role.
For each general election, parties would nominate individual candidates for each lower house district, and a panel of 20 candidates for the senate, including one designated as the presidential candidate. After the national votes are counted, some of the candidates, including the presidential candidate, will be sent by parties as senators depending on their national vote percentages; in this sense, the president is elected by nationwide voting, since at the time the citizens cast their votes, they already know who are the presidential candidates.