Is primogeniture a fair system of inheritance?

Primogeniture is not a fair system of inheritance, and it is not meant to be. It is a way to preserve power.
In primogeniture, the vast bulk of an estate goes to the oldest son in the family on the death of the father (or to the nearest male heir in case there is no son). The idea behind this is not to promote fairness but instead to keep great estates intact. If estates were divided equally between a family's children, we can see how four children inheriting 2,000 acres would each get 500 acres, and if each of them had four children, those children would inherit only 125 acres each: soon a powerful estate would disappear into a cluster of tiny land holdings.
In the eighteenth and nineteenth centuries, primogeniture increasingly came under attack. The highly influential writer Samuel Richardson attacked it as keeping women unfairly dependent in his novel Sir Charles Grandison. Jane Austen likewise shows how primogeniture threatens to throw women into poverty in novels such as Pride and Prejudice.
Some English citizens wanted to settle in the Americas because the system of primogeniture made it very difficult for people who were not born to privilege to acquire land and get ahead in life. Ordinary people usually could only be tenant farmers, renting land from a great lord, in a system set up to enrich the lord and impoverish the farmer.
Today, we can see how primogeniture works by watching the British royal family: the heir to the throne gets the vast bulk of the estates, money, and power in the Windsor family, and the "spares" accept a minor (though in their cases, they still lead very comfortable lives, if at taxpayer expense) portion of the largesse.