Why would I add another power supply?Well, I'll be a monkey's uncle. You found something, and it's RU-Recognized, meaning UL blesses the use of the module as a component of a lamp. So officially, you are modifying your lamp! You know you will also need to add a RU-Recognized 12V power supply of small size (since it only powers the adapter), but it sounds like you can handle that.
The power supply is includedBecause the instructions for the module require it. The module doesn't intake 120V. It intakes 12V. Note the video.