So is the takeaway that universal child care has a positive impact on GDP of the parents, but a negative impact on the overall life of the child?

No, the takeaway is that Quebec's universal child care provision correlates with life issues, but there's no information about the causes of those correlations - which could be in the details of how care is provided and its quality, or some socioeconomic issues specific to Quebec, or for some other reason.

Of course it has positive GDP on the parents: if two women each care for each others children rather than their own, GDP goes up even though the impact on both families is negative (due to lost travel time, as the mothers swap roles).