Here are my conclusions, presented in as a short a form as I as I find possible.
The accepted definition of an Alpha is an almost clinical description of a narcissist/psychopath. Whoever came up with it is projecting his own desires to be worshiped - and be god-like.
The word "alpha" came from the study of wolves and means the parent couple that bred. The pack usually consisted of that couple and their offspring. Transfer that to humans and you get a patriarch and his family.
The commonly-accepted definition of an Alpha (narcissist/psychopath) has traditionally been known as a cad (I like the word "knave" myself), who is a man who lies to woman and tries to manipulate them so he can have sex with as many as possible.
The most well-known bloggers who truly believe in Alphas to Omegas and in the "social-sexual hierarchy" do not have children and families. One well-known blogger is married (or so he says), has no children, has left the United States for Italy, but seems to think he is qualified to speak on relationships in the U.S.
The reason this definition arose is because feminism is about abusing and degrading men and little else. This definition allows men to get away from women, become immune to their abuse, and degrade them so they can no longer be a threat to a man's self-esteem.
Those who are patriarchal try to live their lives according to the Four Cardinal Virtues of Courage (perseverance, fortitude), Justice (giving each his due), Prudence (choosing the best path among many) and Self-Control (not being impulsive). To be more precise, they try to follow all Seven Virtues, with the last three being, Faith, Hope, and Charity (or Love).
To achieve a patriarchal culture it is necessary to women to go along with it. It is in their best interest to do so, because women destroy everything in which they are involved, if too many women are involved.
Alphas/Cads/Narcissists lead their lives the opposite of the patriarchal: they exhibit the Seven Cardinal Sins of wrath, greed, sloth, pride, lust, envy, and gluttony. They lack courage, justice, prudence and self-control.
The Alpha/Narcissist/Cad fears he is cowardly and wimpy, is extraordinary sensitive to criticism (because it damages his weak, fragile self) and fears terribly being humiliated, so he disguises it with braggadocio. (I'll quote James Gilligan, who made the comment, "The most dangerous men on earth are those who are afraid they are wimps.”)
It is patriarchy that advances society and gives men and women the best possible life.
It is the Alpha/Narcissist/Cad who destroys and makes society go backward.
There are those who confuse true confidence in yourself (fortitude and perseverance) with the superficial, manipulative, false charm and false confidence of the conscienceless, loveless Alpha/Narcissist/Cad.
The Greeks believed you achieved eudaimonia (flourishing/well-being) in life through arete, or excellence. You get that excellence through the application of the Seven Cardinal Virtues ("virtue" means "the powers of man"). You do not get it through living a life based on the Seven Cardinal Sins (and "sin" means "missing the mark").
Of course it not either a Cad or a Patriarch. It is a continuum. The more toward the Dark Side, the more everything goes backward. The more toward the Light Side, the better everything gets.