Why use symmetric encryption with Public Key?
I think you're looking for Hybrid cryptosystems.

As you correctly noted it's not only unsafe to encrypt large data with a asymmetric system it's also very slow compared to symmetric systems. That's why we usually use both of these systems together (hence hybrid) to "fill each other's gaps".

We share a symmetric key using asymmetric cryptography. The advantage is that we only have to share the symmetric key once at the beginning of a session and all succeeding messages will be encrypted using the symmetric key. The asymmetric system provides safe key-sharing and the symmetric system guarantees fast encryption / decryption.


  B. Also, if you do have secret symmetric (e.g. AES) keys, you might not want to share them with others, if they have high importance to you.


That's true, but we always create new keys for each communication channel, we don't use a single symmetric key for all communication.