A $1 note being a macro scale physical object enjoys a variety of benefits such as object permanence which provide a baseline level of recoverability. Whereas a wallet key l, being a number, enjoys no such protections.

Of course you may choose to encode your wallet key on paper, metal, or stone granting it properties not unlike a note. However you have now compromised the security of your wallet as well it becomes no mere $1 note, rather it is a note that represents all or a significant fraction of your net worth.