This is a perfect explanation. They could make a national insurance photocard if they'd like. Maybe have police require that you present it within a timeframe. It's the same with driving- you can drive without your licence photocard in the car, you just have to drop into a police station within two weeks if you haven't presented it when asked.

If you even had National Insurance photocards with just the number and a picture, you could just have the employers store them on premise.