The argument is not that "charity improves healthcare" (though in some specific cases it does, locally), it's that since birth rates tend to fall, the claim that "saving a child will only increase the number of starving people in the future" doesn't hold.
I'm afraid that the difference it makes when we decreased infant mortality in a region, is the difference between having 8 children and having 5 children.
The problem is one of culture, education, and relative-wealth (and also whatever traits are infuenced by DNA). Everything else stems from that.
Sub-Sahara population has seen a decrease in infant mortality and at the same time has gone from 250M to 850M people in 30 years.
It's blowing up and I'm also afraid that the cause of all that is foreign intervention and artificial aid. In 2050 there will be 2 Billion people there. Who's going to help feed the hungry then?