Voice mail. It's hard to do this with an app, especially without rooting the phone, since android tries hard to protect the audio stream against malware apps. You could of course do something silly with an external bluetooth audio device.
Someone calls me.
If am not answering within 20 seconds, the app answers the call itself.
It records the meesage.
Once am available, I listen to the messages.
NO Need for operator to record the message.
But as you said, due to restrictions this wont be easy.