Why is there continuity between the leads of my DC power supply?
How do you measure this continuity ? In the diode/beep test of your multimeter ?
Now switch off the supply (very important !) wait a couple of minutes for verything to discharge and now measure again but use the Ohms range.
Is it 0 ohms ? I expect that it is not !

On my Fluke multimeter the continuity test already beeps when I connect 100 ohms.
So a continuity beep does not mean that it actually is a 0 Ohms connection !

Also there will be a lot of decoupling caps on that supply. When they're discharged it looks like a low ohmic connection (to the multimeter) so it beeps.
After some time (could take minutes) the caps are charged and the beeping stops.
Try that with a 100 uF capacitor (or larger) if you that lying around.

So no, it is not bad, just expected behavior ! In fact, this is how I would check that a supply is OK when a device is off, if I would measure 0 Ohms I would check if it might be shorted !