How does the formula for cross sectional area accurately determine which gauge wire should be used without including amperage or voltage?
In Real Life we don't use that formula.  There are tables that tell us what size wire is appropriate for a particular current and environment.

In the US and Canada, the Electrical Code specifies wire sizes for use in AC power wiring.  The American Boat and Yacht Council has wire tables for wiring in boats, with different requirements for engine room wiring, non-engine room, dry vs wet areas, etc.  The type of insulation required can vary depending on the environment.