I think that's only true if you're using direct asymmetric cryptography. In the real world people use hybrid asymmetric cryptography.
Hybrid asymmetric signing is: hash the payload, then use direct asymmetric encryption/signing to encrypt/sign the hash with the private key.
Hybrid asymmetric encryption is: encrypt the payload with symmetric encryption (e.g. AES) with a random key, then encrypt the random key with direct asymmetric encryption using the public key.
As you can see, with hybrid asymmetric cryptography, there's a difference between signing and encryption besides the public vs private key difference.
Hybrid asymmetric signing is: hash the payload, then use direct asymmetric encryption/signing to encrypt/sign the hash with the private key.
Hybrid asymmetric encryption is: encrypt the payload with symmetric encryption (e.g. AES) with a random key, then encrypt the random key with direct asymmetric encryption using the public key.
As you can see, with hybrid asymmetric cryptography, there's a difference between signing and encryption besides the public vs private key difference.