VPN (and other secure communications in general) that are based on both asymmetric and symmetric encryption do not work as you describe.
You do NOT symmetrically encrypt the original data and then asymmetrically encrypt the data again. What you do is to use asymmetric crypto to agree on the symmetric key, and then use the symmetric key to encrypt the data. It is done that way mainly because symmetric ciphers are faster. Much faster. Therefore, you use asymmetric encryption only to send a small data: the symmetric key.
A very simple implementation could be the client chooses a symmetric key by himself, and encrypts it with the public key of the server. Note that the client has the key because he generated it, and the server has the key by using its private key to decrypt it. No one else will be able to decrypt it, because only the server owns the private key. Of course real schemes get a little more complicated to avoid more complex attacks, but this example already mitigates the attack you are describing.
So if you break the asymmetric cipher, you get not the data, but the symmetric key. And with the symmetric key you get the data. But that is fine, since it is not easy to break the asymmetric scheme.