I was poking around on Youtube looking for teardowns of Ebay 12v power supplies and found another teardown where it seems the interior was repurposed from another power supply.
One interesting 'feature' of this recycled power supply is the modification to fit the LED power indicator, it's straight across the filtered output. The connection jack on the other hand is connected
ahead of the filtration choke/capacitors. This means that the LED gets clean filtered power (which it does not need.) and the output to whatever is being powered is effectively unfiltered.
The person who makes the video comments that whoever modified the power supply just attached the output to the "...first capacitor they see ..." (00:11:44 - 00:11:51), while I would agree in essence (If I had modified the power supply I would have unsoldered the original cables and used those connection points for the new connectors...), but there may have been a simple reason for that change, if the output connection and the LED were both connected at the same point, the LED would vary in brightness depending on the current draw.
One good thing about this video is the maker gives a full schematic for what he has torn down, which means it might be possible to replicate the circuit for testing.