I don't believe that the "familial" argument should be interpreted literally in reference to mere biological succession.
The "state as family" is an allegorical concept referring to the duty we all have to serve each other by way of the Sovereign. "The virtues of monarchy," wrote a certain William Stahl (as quoted by Seymour Martin Lipset), "subordinate the individual to the community. Instead of liberty and happiness, loyalty and responsibility are stressed." By swearing to be faithful and bear true allegiance to Queen Elizabeth II, Queen of Canada, her heirs and successors, our politicians are (in theory, at least) swearing to serve a community of hereditary creatures who make up the "family" or "community" we call Canada, not a partisan clique of which they happen to be a member and its policies. The latter route encourages individualistic ambition, which is contrary to the very idea of "family."
A cursory glance at many of the world's republics (both powerful and powerless) demonstrates that individualistic ambition (flowing, perhaps, from "liberty and happiness") is more dear to either the presidents themselves (in the case of executive presidencies, such as France and Russia), or to the parties who put up candidates for purely self-serving political purposes, as was the case in the Republic of India under Indira Gandhi).
"When the Prime Minister bows before the Queen, he bows before us." Alas, the rise of individualistic ambition amongst many of our politicians has succeeded in destroying the symbolism that is implicit in such a statement.