Possibly only women who already have had their firstborn, or are about to do so, would be tempted by this offer. Nonetheless if it helps young mothers find children to be a light burden, then they may very well have more. Other women could see this and so feel encouraged to have children of their own. If this effect snowballs, then the total fertility rate will be increased.
The cost society would have to bear will be large, in terms of government spending. It is likely to cost more to operate a childcare centre than say, the cost to educate a class of primary school students. However, it will probably not be comparable to the total amount of taxpayer dollars spent on education each year, and certainly pales to the defence budget. Fortunately Singapore is currently prosperous, and able to afford it.
There are, however, questions. Why three years? Does it apply to the first 3 children only? What will be the standard of quality? These issues the government will have to look into. Anyway, there is a danger in free childcare. Some people believe that it is still best for mothers to raise their own children, or at least play a big role. By handing over toddlers to childcare centres, some parents may be abdicating their responsibility, and will find it difficult to connect with their children when they are older. Thus while childcare is probably useful to working parents, there has to be a limit so that parents can bond with their children.