You are conflating fairness vs. justice. The world is far from fair -- genetics, place of birth, medical conditions, accidents, etc will always exist like you mention.
The moral/civil question is what is the role of government in an unfair world?
Most governments opt to head down the justice route predominately (e.g. the US's Justice System holding 1/3 of our government powers) while creating a myriad of "fairness" regulations (e.g. welfare programs). The ultimate problem with this approach is people will rarely agree on what is fair. There is no set definition and wants/desires change over time -- even needs do to (like needing a cell phone now vs 20 years ago). This ever evolving definition of fairness is what ensures this debate will rage on indefinitely.