Think about your own history. Your life as an egg actually started in your mother’s developing ovary, before she was born; you were wrapped in your mother’s fetal body as it developed within your grandmother.
After the two of you left Grandma’s womb, you enjoyed the protection of your mother’s prepubescent ovary. Then, sometime between 12 and 50 years after the two of you left your grandmother, you burst forth and were sucked by her fimbriae into the fallopian tube. You glided along the oviduct, surviving happily on the stored nutrients and genetic messages that Mom packed for you.
Then, at some point, your father spent a few minutes close by, but then left. A little while later, you encountered some very odd tiny cells that he had shed. They did not merge with you, or give you any cell membranes or nutrients — just an infinitesimally small packet of DNA, less than one-millionth of your mass.
So Dr. Hampikian is trying to stress how little the father is involved in the biological history of any given child in comparison to the mother. The father's contribution is miniscule and to the extent that it is necessary it is only marginally so. But in drawing out his narrative he is either intentionally or accidentally being very confusing.
You are not your mother's egg. You did not begin to exist when her eggs were formed. Her egg is a part of her body with a very specific reproductive purpose. From Robert George and Christopher Tollefsen's Embryo:
Is either the sperm or oocyte... identical to Smith, the adult whose origins we wish to specify?
The answer to all these questions is negative. Sperm and egg cells are parts of the human organism... a part of a biological organism in the sense that concerns us here is some living subset of the cells that comprise the totality of the organism , the life of which subset is integrated into the life of the whole, and which performs some unified functional role within the life of the organism.
They point out that both the sperm and egg after being released to serve their reproductive function will quickly cease to exist if fertilization fails to occur. Our ability to use scientific advancement to artificially preserve the life of the parts does not change their nature. They are meant for a specific purpose within the organism they help comprise and once they serve that purpose or fail to serve that purpose they quickly cease to exist.
The zygote on the other hand is entirely different. Quoting Keith L.Moore and T.V.N Persaud in The Developing Human, "Human development begins at fertilization when a male gamete or sperm (spermatozoan) unites with a female gamete or oocyte (ovum) to produce a single cell - a zygote. This highly specialized, totipotent cell marked the beginning of each of us as a unique individual."
It is easy when telling the narrative of our worldview to hedge the story a bit to make our position seem more obviously true. In Dr. Hampikian's desire to highlight an obvious truth, that women are more biologically invested in the reproductive process than men, he sails past telling the story into redefining the nature of our life. We were never an egg that met up with some sperm after having sustained existence for years upon years. Without both the sperm and the egg our story never begins at all. I understand that this fact somewhat undermines the overall point Dr. Hampikian is trying to make, but it is a biological fact whether convenient or not.
I will leave it others to address the idea that men are not necessary after conception. It is not a tall hill to climb as you have the support of mountains of evidence that intact traditional families contribute to the well being of children in numerous and often surprising ways. It is enough that in order to minimize the impact of men in sexual reproduction he felt it necessary to reduce us all to components of our mother's reproductive system and - in doing that - blurred the lines between parts of human beings and the whole distinct living human beings we all are.