For messages sent through an insecure channel, a properly implemented digital signature gives the receiver reason to believe the message was sent by the claimed sender. While using a digital signature

Encrypting with the sender's public key would mean anyone could create the signature, completely defeating the purpose of proving sender identity.

The receiver's private key is used for decrypting messages sent to them, not for verifying a sender's digital signature.

Encrypting the digest with the receiver's public key would make it a confidentiality operation (only the receiver can decrypt), not a signature that proves sender identity.

The sender uses their private key to encrypt the message digest (hash), producing the digital signature. Because only the sender possesses their private key, this operation provides non-repudiation - proving the message originated from the claimed sender. The receiver decrypts the signature using the sender's publicly available public key and compares the result to a freshly computed hash of the received message to verify both authenticity and integrity.