What I don´t understand is how it can be the cause of it, because in theory the diminution of infantile mortality is the main factor and has to come first.
Anyway, France had his demographic boost - lot of active people in the population, less dependant one with less children and not many elders yet - at the end of 18th /beginning of the 19th and squandered it through the revolutionnary and napoleonic wars, and was a bit too early - already stabilized - for riding the industrial revolution.
The third republic obsession with natality was real, and can be easily understood: if for 30 years you were an outlier with less natality, were crushed by expanding neighbours and could not conceive that they will slow their natality one day too, you could have felt uneasy... La répartie est dans l'escalier. Elle revient de suite.