When the mayor of the town confirms the number of illegals he is happy to have working and the French consulate gives figures for the total I have no doubt the journalist was doing his job. The government collects the tax whether the SS number its attributed to exists or not.
The false driving licence thing is used by some Europeans. A local biker presents a false US licence when stopped to avoid points - he only has to pay the fine. The Americans are incapable of telling their own beat cops if a licence is false let alone the French police.
They interviewed a series of illegal immigrants who had been living and working in NY on false papers for years. A musician, a danser, a barman, a babysitter, a restaurant worker.
You don't have to believe the Earth is (more or less) round, doubters. You'll be telling people from Iraq and the 'stan they can't live off the state in the UK next.