It's as simple as this:

This power supply is not capable to deliver enough turn-on-current, which usually is higher than the continuous current needed for idle run.

Even if it would, it would be too weak to deliver enough power when ADI-2's  headphone amp is used (2A / 24W max).

24 W is really not a big deal for a power supply.
If you really want to spend that much money get yourself a medical grade PSU with reduced ground coupling and mains leakage.

The LPS 1.2 claims to do that, and I 100% support the necessity of such, if you have unbalanced interconnects and want to keep the full dynamic range capabilities of the DA converter.
For balanced interconnects it doesn't matter.
This is not audiophile BS.

In so far it's a bit sad that this PSU does not work with the ADI-2 DAC.

I would call this a design flaw on the PSU to limit current to just about spec's with no margin, as it's almost impossible to avoid higher switch-on currents, specially with small devices where additional soft start circuits are overkill to implement.

Here is a lengthy quote from Uptown Audio:

The most important result of the isolation afforded by the UltraCap™ LPS-1.2 is that it completely blocks the path of AC LEAKAGE CURRENTS.  Some more in-depth information on the subject can be found on the Swenson Explains page, and here is bit more condensed explanation of this important topic—also written as a forum post by John Swenson:

The PRIMARY purpose of the LPS-1.2 is to break leakage current induced ground loops, (which exist in almost all systems), and the fact that the output is isolated from the input (thus not affected by noise of the feeder supply) is a bonus, not a primary goal. Of course the LPS-1 has a very low noise, very low impedance output as well.
There are two very different mechanisms by which an AC power supply can interact through the AC mains:
1) traditional "noise injected back into the AC line." This is easy to measure, easy to grasp what is happening, and many devices exist to try and filter out this sort of noise.

2) Leakage current, this is created by capacitances between the AC line and the DC output of the supply, it ALWAYS exists but varies from supply to supply. This is hard to measure, hard to grasp what it is and how it propagates through a system. After everything else is taken care of the leakage current is still there, I consider it to be one of the last great untouched detriments to obtaining best sound.

Both #1 and #2 exist in almost any system to varying degrees. Since there is already a large body of knowledge and products to deal with #1, I set out to work on finding ways to deal with #2. The LPS-1 is the first fruit of that work.

Leakage current has always been there, but has not been a big issue until recently for two reasons:
A) it is fairly low level in most systems so its effects have been masked by all the other problems preventing our systems from being their best, as these other sources of problems have been identified and addressed, the sonic results of the leakage current are now much more obvious.
B) leakage current from SMPS is usually much higher than from LPS, so the inclusion of computers (which are usually run from SMPS) into systems has greatly exacerbated the sonic detriment due to leakage current.