I do not see why you would be sending this graph. In the end, GPG (and subsequently, the way PGP is designed to) creates an encrypted file that can be decrypted by only the private key of the recipient unless you use symmetric encryption. The graph you sent is different than what you would usually refer to symmetric file encryption and in the end, you need a private key to decrypt that file and not a passphrase that is the same as the one that was used to encrypt that file.
and yes, this is exactly what symmetric is. You use only 1 key to encrypt and decrypt.
--symmetric
-c Encrypt with a symmetric cipher using a passphrase. The default sym-
metric cipher used is AES-128, but may be chosen with the --cipher-algo
option. This command may be combined with --sign (for a signed and sym-
metrically encrypted message), --encrypt (for a message that may be
decrypted via a secret key or a passphrase), or --sign and --encrypt
together (for a signed message that may be decrypted via a secret key or
a passphrase). gpg caches the passphrase used for symmetric encryption
so that a decrypt operation may not require that the user needs to enter
the passphrase. The option --no-symkey-cache can be used to disable
this feature.
And as you know, aes-128 is a symmetric encryption algorithm. However when you use GPG to encrypt to a public key, you need the corresponding private key. I don’t see where the problem is to be honest. It clearly says it does symmetric encryption in the manual and it uses symmetric encryption algorithms