It's not even about sterilization, but about creating such unfavorable conditions that women are forced to leave or give up the child until better times come.
Exactly, if the women see no future there then they clearly won't seek to bring children there.
Women at the time didn't have much choice. The mass of peasantry were serfs, owned by the lords, included as amenities on the land. The fact that fertility was higher during famine than during the good economic times today suggests that the thoughts of women on future prosperity have nothing to do with fertility rates.
The decline of testosterone in men to a fraction of what men in the 1930s had is a far more likely factor. No one has a choice in that, when availed products of industrial production, apparently.