Letting terminally ill people die will do little to slow population growth because the vast majority of these people are not going to have (additional) children anyway. But there are other solutions that would actually work.
You write that there seems to be no slowing of procreation. This is quite false. Total fertility rates (average number of children per woman) have fallen spectacularly since 1955 ... but only in countries and regions where poverty has been meaningfully reduced. For example, the TFR of East Asia fell from 5.42 to 1.72 (below the rate of reproduction) -- and East Asia is the most populous region on Earth. So this is highly significant. There were large drops also in Portugal, Australia, Botswana, Italy, and so on. Where poverty persists, on the other hand, so do high TFRs. Many African states are good examples of this. Niger's TFR has increased from 6.86 to 7.15, and other high-poverty countries (Mali, Senegal, Equatorial Guinea) are not far behind. The evidence is overwhelming, and the lesson is clear: ensure that people have some social security, so they need not depend in their old age on having surviving children; ensure that infant mortality is low, so people need not get lots of children to ensure that a few will survive; ensure that women have good educational and employment opportunities so that they have important social roles other than that of bearing and raising children.