Less than two weeks after Larry "Major Nelson" Hryb said he would look into why it isn't possible to get a replacement power brick through the official support site, he has now let it be known in the same thread on Reddit, that this is now possible.
Hey gang, I have an update on this. I have worked w/ the Support team and we've enabled the option to purchase additional Xbox One power supply units via https://myservice.xbox.com
I have not had a chance to check it out myself yet - but the team just emailed me to let me know they have added that option to the online tool.
Xbox One owners will now be able to use the Online Service Center to replace a faulty power brick free of charge, providing the unit is still under warranty, or purchase additional ones, rather than go through a third party vendor. However, the note on troubleshooting power related issues does state that the console must be registered:
You must register your product online to replace the unit under warranty. (The power supply warranty is the same as your console warranty.)
Thanks for the tip Jack Wilkinson!