While I generally agree with your conclusions, I have to take issue with this statement.
A matriarchal society does not equate to a fatherless family. A matriarchal society is one in which the female takes the leading role in the decisions made about family affairs, not one in which there are no males. A society with a large number of female single-parent households is not matriarchal nor is a society with a large number of male single-parent households considered to be patriarchal. There is a word for single-parent households. The word is disfunctional.
Traditional Hebrew families and many Asian cultures are quite matriarchal yet don't have the same problem as American single-parent families do. It is not a relevant comparison.
If you meant to blame the destruction of the nuclear family for the problem, I will agree.
By your definition that, A matriarchal society is one in which the female takes the leading role in the decisions made about family affairs , what goes on in the inner city seems to fit. Actually, I did not think of this example myself. I recall Walter Williams talking about it on the radio.
There are other examples of matriarchal societies that have worked better (as you have pointed out): some aborigines, some aspects of the Hebrew culture...
There are numerous examples of patriarchal that have or haven't work well, also.