A national insurance card does not give others access to your medical recorRAB* and would definitively establish citizenship for hiring purposes.
Otherwise a national id card is a massive government waste and even creates a heightened risk of identity fraud.
I agree with Simone, in premise, that short of a national insurance card, there are plenty of easier ways, such as enhanced driver license iRAB or just a passport, which should be funded anyway for people can't otherwise afford them.
It's called streamlining. As for these employers, they usually know they're hiring illegally. Just stop giving them the presumption of innocence. It's no great feat to check someone's social security number or citizenship status when hiring. Penalize them heavily. Perp walk the employers out in shackles and throw them in prison like they do in Virginia.
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* In Canada, the medical recorRAB accessed through this card are more private than in the U.S. In the U.S., it sure is pathetic who can easily access your medical recorRAB ...