However, if you press any buttons on the unit it wakes back up without powering up the amp so basically the player is fully functioning with the key off without any sound.

This could be related to your other problem, or it could simply be bad wiring in the installation.

The ability for the player to wake up after the key is removed is a feature. It's detailed in the FAQ. You are supposed to be able to wake the player up from Standby mode if you want to.

If your player is wired correctly, this will also bring the Amp Remote wire high and will activate your amplifier(s). However, if your amps or your player are not properly wired (for instance, if they are connected to the ignition wire which would be a very Bad Thing), then the behavior you describe might happen.