I think it's rather more than the fact that they alone had members of their Legion turn traitor. Right off the top of my head, for instance, one of the Garro audio dramas has traitor White Scars. I would be shocked if they were unique in this.
The larger issue, I believe, is that the time after the Heresy was defined by paranoia and an attempt to maintain a delicate balance of power. Dorn and Guilliman almost went to war, and there was a deliberate effort to take away power from those that couldn't be trusted.
During this time period, the Dark Angels had lost their Primarch and had been forced to destroy (for all intents and purposes) their home planet. This, after their Legion's nominal second-in-command had turned traitor and subverted Caliban to create his own mini-Legion*.
In this sense, one can understand why the Dark Angels were keen to keep a secret. They might have had skeletons in their closet, but they had no reason to fear anyone discovering them. That changed when they discovered the Fallen still existed.
The Codex reinforces that, initially, the Chapters that would become the Unforgiven were among the hardest-fighting forces of the Imperium. But that would have meant nothing in M31 if the Fallen were discovered by outsiders. Would the nascent Inquisition and High Lords have accepted the Dark Angels' lies as justified given the purges and infighting going on at the time? Who knows. The Dark Angels don't seem to have thought that would be the case.
The problem is that, since then, the Dark Angels have simply upped the ante where keeping the secret safe is concerned. Any chance of being let off has sailed, given their litany of sins in pursuing the Fallen. The Imperium would not destroy the Unforgiven, of course, but they would put them on some sort of horrific Penitent Crusade, and they would certainly swamp them with Inquisitorial overseers.
* In this sense, it's actually kind of a misrepresentation to assert that "half the Legion" (a number that has no real basis) turned traitor. Point of fact, Luther was training recruits who were shipped off to the Legion. When he decided to turn traitor, he might have had a handful of Chapters besides his initial training cadre of five hundred Astartes. Either way, once he turned on the Lion, it's likely that none of his warriors considered themselves Dark Angels - or, at any rate, a part of the Legion (since the Legion would have been representative of the Imperium they were rebelling against).