Population will also drop if there is a significant fall in prosperity. Fertility rates may rise, but fatality rates will rise much faster. Populations do not tend to rise when people are starving to death.
On the other hand, a sudden increase in prosperity will reduce mortality much faster than it changes reproductive habits. Posterity under those conditions will descend from those who lack prosperity. The more successful economically will be less successful in the Darwinian sense.
FIRST: Your statement does not affect the fact that these billionaires are acting rationally when attempting to reduce population growth by means of greater prosperity, etc. After all, they could hardly call for increased famine and civil unrest as a solution to alleged overpopulation.
On the whole, their logic is reasonable. (I make NO STATEMENT about my opinion on its political aspects.)
SECOND: Your statement is valid only given EXTREME (Armageddon-like) conditions - conditions not found in the bulk of the countries in which these billionaires might wish to enact their policies or launch their programs.
On the other hand, a sudden increase in prosperity will reduce mortality much faster than it changes reproductive habits.
I believe that this is pure speculation on your part. I namely don't believe that there are any historical precedents for it in recent times, in "normal" countries. On the contrary, evidence indicates that reproductive patterns and habits can change very quickly (resulting in fewer offspring) when economic conditions improve. It is impossible for me to debate your statement's validity without a more-rigorous formulation and definition of terms (like "sudden").
Regards,