I don't think the data support the conclusion that contraceptives are the main driver behind the decline in fertility. Contraceptives have been available for a long time and fertility is continually decreasing even when nothing changes in the availability of contraceptives.

Also, data going further back show that fertility has been declining since before contraceptives were widely available. Sweden for example: https://www.statista.com/statistics/1033535/fertility-rate-s...