> They can be created for any purpose and people can trade solely on the face value of the check without actually presenting them to the payee bank.

The system is still dependent on a centralised trusted authority (the bank) verifying transactions. Deferring that verification doesn't mean it's no longer required. And in practice no one (other than maybe loan sharks) actually does so.

Crypto is trustless. No central authority is required to verify anything.

> Notes are very similar, only there's no third party drawee. People used to trade in notes prior to the Civil War and still do. The dollar is a type of note.

No. Notes in the original sense were backed by a commodity. The dollar has not had such backing since Reagan.