Families are much closer to autocracies than to liberal democracies. Parents have incredible control over the lives and activities of their children.
Some families may aspire to a more liberal constitution, but even there the reality will only allow for so much freedom, especially when children are in their younger years. Allowing a judicious dose of freedom whenever possible can certainly be a good and commendable thing, but such freedom must always have limits, imposed by the parents, and even valid uses of freedom may sometimes necessitate attempts at persuading the children to use their freedom more wisely or beneficially.
To whatever extent a family will be a dictatorship, it can be either a benevolent dictatorship or a tyranny. The former will seek the good of the governed, and the good of the whole family. The latter will use its power selfishly, trying to maximize benefits (or more likely, minimize inconveniences) to the rulers.
Even the most liberal of states will in this way be absolutely filled with illiberal activity, of a (generally) positive sort. An honest defender of liberalism will admit this fact without concern, though someone too consumed by ideology could try to downplay this obvious truth.
The family is the most natural and appropriate place for such illiberal government. The level of government that is furthest removed from the family or the individual is the place where it is least suitable.
And yet, this does not mean that any level of government should shirk its duty, should try to pass the buck to lower levels. It is inappropriate for a higher level of government to try to micromanage the activities of individual citizens, as a parent might have to micromanage the activities of an infant. However, where the higher levels can make changes that benefit all citizens, it is appropriate for them to do so, rather than hoping that lower levels will pick up their slack.
In some ways, modern politics is the worst of both worlds. The politicians are as self-interested as any tyrant, and the state is permissive of immoral and destructive behaviours in the name of freedom. I think it would be possible to keep many of the strengths of liberal democracy without needing to hold onto these worst qualities.