If you add some dancing with chicken feet and candles to your NFP scheme - the Haitians might try it for a day. Until they lose interest and go back to being low-IQ adult children.
Despite all that, less than 40% of Haitian women use any contraceptives at all. Very obviously, only coercive methods would be "successful" as some seem to define "success," which means suppressing Haitian childbearing without reference to the overall health, dignity, or even choice of the people involved.
I trust you'll be intelligently curious about people who have actually implemented fertility-awareness-based NFP programs. This article references the inauguration of a NFP program in Haiti
... and this one takes a larger view of contraception and Third World poverty.
Here's a good one on NFP in India, by the British Medical Journal.
The following from the Abstract is interesting:
Natural family planning, when used by motivated couples, is a safe and cost-effective means of birth control. ... A total of 869 women of diverse ethnic and economic backgrounds participated in a study conducted by the World Health Organization. Regardless of literacy and culture, 93% of the women were able to recognize the changes in their cervical mucus associated with ovulation... A failure rate of 0.2 pregnancies per 100 women was found in a study of 19,843 women in India..