Yes, I think that is a fair summary. The only thing I would say is that I don't think Engles is using 'gentile' to mean non Jew. He appears to be using it to mean 'blood relatives on the mother’s side'.
There is quite a good summary here - the relevant section is 'The Marxist method' which is worth reading in its entirely but the following quotes give you the bones of it.
As production shifted away from the household, the role of reproduction changed substantially. The shift toward agricultural production sharply increased the productivity of labor. This, in turn, increased the demand for labor–the greater the number of field workers, the higher the surplus. Thus, unlike hunter-gatherer societies, which sought to limit the number of offspring, agricultural societies sought to maximize women’s reproductive potential, so the family would have more children to help out in the fields. Therefore, at the same time that men were playing an increasingly exclusive role in production, women were required to play a much more central role in reproduction.
The rigid sexual division of labor remained the same, but production shifted away from the household. The family no longer served anything but a reproductive function–as such, it became an economic unit of consumption. Women became trapped within their individual families, as the reproducers of society–cut off from production. These changes took place first among the property-owning families, the first ruling class. But eventually, the nuclear family became an economic unit of society as a whole.
The nuclear family: the root of women’s oppression.
It was under these circumstances that the monogamous nuclear family–the family as we know it–began to take form. The modern family arose for one purpose only: to pass on private property in the form of inheritance from one generation to the next. All of the romantic imagery of "true love" which has since helped to idealize marriage in contemporary society can’t change the fact that marriage is essentially a property relationship. Most people learn this all too clearly if they find themselves in divorce court.
The Original meaning of the word "family" (familia) is not the compound of sentimentality and domestic strife which forms the ideal of the present-day philistine; among the Romans it did not at first even refer to the married pair and their children but only to the slaves. Famulus means domestic slave, and familia is the total number of slaves belonging to one man. As late as the time of Gaius, the familia, id est patrimonium (family, that is the patrimony, the inheritance) was bequeathed by will. The term was invented by the Romans to denote a new social organism whose head ruled over wife and children and a number of slaves, and was invested under Roman paternal power with rights of life and death over them all.
But there was a further contradiction between earlier communal social organization and rising class society, Engels argues. Wealth was owned by men, but since most societies were matrilineal, inheritance was passed through the mother, not the father. Moreover, without strict monogamy, a man cannot be certain that his wife’s children are also his own. Engels writes,
Thus, on the one hand, in proportion as wealth increased it made the man’s position in the family more important than the woman’s, and on the other hand created an impulse to exploit this strengthened position in order to overthrow, in favor of his children, the traditional order of inheritance... Mother right, therefore, had to be overthrown, and overthrown it was."
But whether or not all early societies were matrilineal is not as important as it might seem. What is indisputable is that the onset of class society brought with it a universal shift toward patrilineage–and, more importantly, the role of men as "heads" of their households. Engels was undoubtedly correct–with more supporting evidence today than when he was writing–that the rise of the nuclear family brought with it a degradation of women which was unknown in pre-class societies.