ReadyNAS Pro 6 won't start up

It sounds likely to me that the power supply has gone bad.  The LAN ports shouldn't be "flickering".  The LAN ports and the power-on circuit both use a part of the supply labeled "+5VSB", which stands for +5 Volts Standby.  That voltage stays on even when the NAS is off, so is often the first to go.  And the whole NAS will look dead if it's dead.

The Pro6 uses a standard SFX format supply, though the 24-pin connector cable is typically too short on a replacement, and there may also be too few 4-pin Molex conenctors (like an old IDE hard drive used) for the SATA backplane (it needs 3).  While replacement isn't really "trivial", it's something most can do if they take care.  A 6" or so ATX extender and one or more SATA to Molex adapters may be needed.  Or, you can move the 90° Molex from the original to the new one since they are insulation displacement style.

The orignal Seasonic supply is no longer manufactured, but there is a seller on eBay who sells supplies already modified to be a direct replacement if gathering the supply and adapters separately isn't your thing.  If you have a spare ATX supply available, you can plug that in externally just to verify a new supply is the answer before you invest in one.  If your external supply doesn't have at least 2 Molex connectors for the backplane, just remove your drives when you try it.