Not to defend the social security number system (because it’s terrible for unique identification of individuals and when it was created it wasn’t even intended to be used for that purpose) but you’re not supposed to carry it around with you all the time. It’s supposed to be in a drawer/safe/fireproof bag with the rest of your important documents, and only brought out when you need it.
Why make it a card then? If it’s meant to be stored as a document then it should be the size of an envelope, or a standard A4 sheet.
I keep mine attached to the envelope sized paper it came on. But it shouldn’t be a card shape, that encourages people to carry it. My grandpa always carried his.
Don’t mention A4 paper, it scares the Americans. If any of them could read, they would be very afraid right now.
