Perhaps if we started with some definitions.

A public good, as I understand it, is one that can be
consumed by an individual without its being destroyed
and so not available to the rest of society.  Most
certainly children can be destroyed in use.

Human capital, like all capital is not totally consumed
in use, but some use is left for a future time.  Some
human capital may be augmented by use.  Having and
raising children is investment in human capital.  Most
certainly the investment, at least in part, can be
destructive.

A positive externality is something like a public good
and human capital.  One agent uses it, but in the
process a good is generated for other parties.  Most
certainly, however, there are negative externalities.

Into which of these do children fall?

Then we must consider that while some  couples
want children other couples acquire them
as an unintended consequence.  Further, it
is not until the child is (say) seven years old that
it is evident that they are going to be a good for
society or a bad.

In the light of these considerations I wonder if
children can be fitted into the limited categories
of micro economics.

Robin Neill