'Cron Souls

That would actually make sense. The Old Ones did use energies beyond Necrontyr comprehension, after all, but there's evidence the Cron had emotions. As far as I can tell, that requires a soul.

I don't believe so, but a lot of peeps seem to insist that it was the souls that were eaten despite the wording

The warp has previously been described as anathema to the C'tan, at least in the 3rd ed codex. I don't recall the Warddex contradicting it.
Where does this prominent theory come from?

Given how much it seems the C'tan wanted to seperate the material universe from the warp, and how they appeared to loath/fear psykers, that doesn't really make any sense. Helping to either develop or cultivate more blanks in humans came across as a counter strategy against psychic races, with Pariahs being a sort of culmination of millennia long plans. But that's all been retconned now.

It wasn't souls in the beginning, pic related. But I understand that the FFG material somehow muddied that, so who knows now.

6e Necron codex in the "In the Beginning" chapter. Specifically calls out eating souls and the C'tan being the anathema to the Old Ones.

Prominent as in I have heard the theory from multiple people irl and from multiple websites (not just this one). It may not be the actual prominent theory, I just know that I have heard it multiple times over a course of a few years.

The 6ed codex mentions a prior war between the C'tan and the old ones, and that the C'tan had been defeated and so hidden themselves away. I cannot find a reference to the C'tan being regarded as anathema to the old ones anywhere there.

In the Biotransference part it describes the C'tan swarming around the sites where the Necrontyr conversion took place and that they consumed cast off life energy/essence.
It might be open to interpretation whether that counts as a soul and if that accounts for what the Silent King regards as the loss of the Necrontyr race's spirit, but the distinction between the two seems apparent in choice of words there.

This will probably read as my own spin on events, but the soulless revelation the Silent King comes to is kind of clear given what we know of Necrontyr technology from both codices:
Necrontyr had always been perplexed by warp based phenomena, which is reflected by their constant defeat by the old ones during the pre C'tan stage of conflict. I doubt their science could correctly transfer (or even quantify/acknowledge the existence) of a soul. When the transference happened the Necrontyr truly died, their souls lost forever. The process was brilliant, their greatest minds encoded into undying ageless bodies. But ultimately flawed in that one regard.

Xenos don't have souls.

Report to the nearest Priest for deprogramming, heretic.

...permission to request a data expunge, partial controlled memory wipe and surrender of heretical material?

Permitted. Don't re-offend, citizen.

But the Old Ones lost at all parts of the War in Heaven...

I'm actually not so convinced...I have a running hypothesis that there's something a little more...elegent, happening in the background there.