The specific wording of a law is not quite the same as a courts interpretation, and enforcement's instructions.
In other words, if a law says X as you and I read it, but a court says that X actually means Y, and enforcement is based on compliance with Y then it can pass muster, until./unless a different court overrules the first one.
FOID cards are not new things. Several states have had them or some equivalent for decades. I would think that if there was something grossly and clearly unconstitutional about the concept it would have been challenged in court a long time ago.
But, perhaps not, It took literally decades before a valid challenge to DC's handgun ban finally made its way through the court systems.
I wouldn't worry over much about being prosecuted for not having your card physically on you 24/7 in your own home. While the language of the law may seem to allow it, the practical side of things is that, generally speaking, if you don't have the gun physically on you (or in your hands) you don't have to have the card on you. If you do have a gun on you, then, you do, or you could be charged.
I'd love to see a case where you were grabbed from your bed in the middle of the night, and charged for not having your "papers" on you, for a gun locked in a safe in a different room of your own home. No prosecutor would be stupid enough to try and take that before a jury. And even if they did, even the most gun hating judge would have a hard time not throwing that case out.
If they make the argument that you had "possession" of a the gun because it was in your house, under your control, then you ALSO have possession of the required FOID in your house and under your control at the very same time.