My plan was to use asymmetric encryption where the secret key is again encrypted using something like AES

I think your terminology is off. AES is an example of symmetric encryption: Decryption requires the same key as encryption.

An example of asymmetric encryption would be public-key cryptography: You encrypt a message with the public key, but only a private key can decrypt the result.

AES should be fine for encrypting large blocks of data.

I believe that for systems like TLS, asymmetric encryption is only used briefly to negotiate a symmetric key between client and server.