Based on quite a bit of research into this area it seems that 'fertility awareness methods' can be very effective if not on par with many other popular contraceptive methods:
https://www.ncbi.nlm.nih.gov/pubmed/17314078
https://www.ncbi.nlm.nih.gov/pubmed/17371516
https://www.ncbi.nlm.nih.gov/pubmed/19264938
Clearly, this method is best used within a long-term monogamous relationship as it does not protect against STIs.
Are there other reasons why it is not widely-known and promoted by physicians given the fact that there are no side effects for both partners?