"In fact, the smaller 12-volt battery used to start the gas engine and run accessories is more prone to failure."
The 12-volt battery does not start the engine. The engine is started by spinning one of the high-voltage motor/generators, to spin the engine up to regular engine idle speed (around 1,000rpm). The car doesn't have a traditional 12V DC starter motor - if the HV battery is dead, the car cannot be started.
The high-voltage battery is normally kept isolated - disconnected from the power electronics under the hood. The 12V battery must have enough charge to boot all the computers (requiring about 30 amps of current), and then to turn on the System Main Relays which connect the HV battery to the rest of the system. Once the HV battery is connected the high-voltage components take over, and can recharge the 12V auxiliary battery from the high-voltage system, even if the engine isn't running yet.
Because the 12V battery has a relatively low capacity, leaving the car in 'Ignition On' (where all the idiot lights light up, by pressing the Power button twice without pressing the brake pedal) will drain it pretty quickly - under an hour.