Just because your batteries are reading 12+ volts does not mean the amps are getting to your starter to turn the engine over. It sounds to me like you may have corroded connections at the battery or the generator end because your generator turns over with the eng running. To know for sure I would connect a battery or battery from your tow car directly to the generator. If it starts you have a problem in the cables from your house battery to the genset. If it does not start then I would look at the starter.