That's not the case. SIM cards hold the permanent key for authentication and perform key derivation. Mobile data doesn't pass the SIM card; it does not perform the encryption and decryption.
Dumb/feature phones saved SMS messages to the SIM card as simple cards have a limited amount of memory that is dedicated to a crude phonebook and SMS store.
Smartphones and smarter feature phones (can) use their own storage for that. You could disable/enable the phonebook/save to SIM features on feature phones and early smartphones.
(I'm talking about win CE and symbian phones being early smartphones here)