Most public/private use symmetric encryption for the data anyway but they use a key derivation scheme to share the key. You can't use public/private encryption

Most public/private use symmetric encryption for the data anyway but they use a key derivation scheme to share the key. You can't use public/private encryption for data size greater than the key size. Obviously you can sign/verify any data size. Also, above you said "encrypt this symmetric key with your private key", did you mean sign this symmetric key?