That's exactly what a check is. It's a legal document that says "I give you X dollars from account Y." You can use a napkin instead of a the pre-filled sheets of paper your bank sends you, and it's still the same legal document.

Typically your signature is checked against one on file, but only for large transactions, and of course handwriting signatures can be forged. And, if you are writing checks against other people's bank account, you'll probably go to prison. That is the check on the system.

(It's no different than signing a contract. I'll do this work and you'll pay me when it's done. What happens if they don't pay you? You sue them. The root of trust is the judicial system.)