The idea is that if it gets lost outside somewhere, then the paper will degrade to the point that it's no longer readable. You can always get another card if you lose yours.
When a number holder (NH) reaches his or her yearly or lifetime SSN card limits, an SSN replacement card may still be issued if failure to receive an SSN card will cause a hardship.
To receive an exception due to hardship, the NH must present a letter from a third-party provider, such as a State public assistance agency or an employer, indicating that the NH must present his or her SSN card to receive employment or a benefit.
So you're not completely fucked, it's just some extra leg work.
It is, but you don't need the physical card for very much. I've only ever needed the card for jobs. I've never had to apply for welfare, but I do belive you need a physical card to apply. But in most situations, you just need the number.