RN426E22 NAS Stuck on "Welcome to Netgear Storage" – No Button Response, No Network Access

On an RN628X, I have confirmed that the default display is ReadyNAS Storage.  Welcome.  So I'm now even more sure what you are seeing is just the default display.  The NAS does not even have to complete POST (Power-On Self Test) for that to display.  I have no easy way to check, but I'm pretty sure it runs off the 5V rail and may not even require especially well-regulated power.  So, as I said, there are many potential causes of that symptom.

In addition to posting the result of the diskless test:  How did you turn it on?  Did you have to press the power button, or did it just come on when plugged in?  When you turn it on, do you even see a flicker on the Ethernet LED?   Do the drives spin up?  You say you get no results from pressing any buttons.  Does that include a long press of the power button?

If it is the power supply, then it's likely either the +3.3V rail (unless the display runs off that, which would be unusual) or the +5VSB (5 V standby) power.  But it could be a weak +5V or +12V rail.