There's not really any discrepancies in the theory, per se, but it is one of many theories introduced by modern scholars trying to fill a discrepancy in the surviving sources.

The theory that Cyrus the Great/II was not an Achaemenid has been a mainstream topic of debate in Achaemenid Studies almost since the decipherment of the first cuneiform texts from within the Persian Empire. In the last few decades, it has gained steam as a near consensus among historians. That debate stems from two primary documents: The Cyrus Cylinder and The Behistun Inscription.

Unearthed in the ruins of Babylon by the Assyrian archaeologist Hormuzd Rassam in 1879, the Cyrus Cylinder is one of extremely few contemporary sources from the life of Cyrus the Great. It is primarily a proclamation of Cyrus' legitimacy to the people of Babylonia, written in Babylonian Akkadian either by or in consultation with the priests of Marduk, the Babylonian patron deity. One of the many ways that Cyrus portrayed himself as a legitimate ruler in the Cylinder text is by explaining his royal titles and heritage:

I am Cyrus, king of the world, great king, mighty king, king of Babylon, king of Sumer and Akkad, king of the four quarters of the world, the son of Cambyses, great king, king of Anšan, grandson of Cyrus, great king, king of Anšan, descendant of Teispes, great king, king of Anšan, of an eternal line of kingship... (A.20-22)

That is the only contemporary genealogy of Cyrus the Great (or his own son, Cambyses II), and while similar to the genealogy presented by later kings, there is one important difference. This list of ancestors stops at Teispes.

Roughly 20 years later, after seizing the throne in a coup, Darius the Great/I commissioned the Behistun Inscription. Inscribed in Akkadian, Elamite, and Old Persian on the side of a mountain, it explains the circumstances that made Darius a legitimate king, at least as he wanted to present them, and his victories in the civil war that followed his ascension. Once again, part of that legitimacy was through a presentation of his genealogy:

King Darius says: My father is Hystaspes; the father of Hystaspes was Arsames; the father of Arsames was Ariaramnes; the father of Ariaramnes was Teispes; the father of Teispes was Achaemenes.King Darius says: That is why we are called Achaemenids; from antiquity we have been noble; from antiquity has our dynasty been royal. (1.2-3)

So here we have a genealogy that traces Darius' lineage back to Teispes as well, but then continues on to Achaemenes before beginning what would become a long tradition of stressing the family name "Achaemenid" as part of the royal titles. It is generally understood based on the extant documents and a line later in the Behistun Inscription itself that the Old Persian cuneiform script was invented specifically to create this monument with the Persian language, which means that the two Old Persian inscriptions at Cyrus' palace of Pasargadae identifying him as "Cyrus, an Achaemenid" must also have come after Darius was king. This is not a terribly controversial stance since inscriptions were often some of the last things added to construction projects and it is known that Darius was king by the time the whole palace complex was completed. All of the Classical stories take it for granted that Cyrus was an Achaemenid.

The problem is, historians are sure that Darius was lying about parts of his story and legitimacy, but reaching a consensus about exactly what he was lying about is difficult, especially because absolutely all of the Classical sources reported his story more-or-less as presented at Behistun. Darius claimed that Cambyses II had murdered his brother Bardiya, the second son of Cyrus the Great, and had him replaced by the magus Gaumata. Gaumata, still claiming to be Bardiya, usurped the throne while Cambyses was in Egypt, but before he could return, "Cambyses died his own death." Various Greek sources reported that death as something to do with an infection stemming from the king accidentally cutting himself with a knife. Thus Darius and six co-conspirators infiltrated the palace and murdered this magi pretending to Bardiya, proclaimed Darius king, and fought a civil war against at least five other rebels claiming false identities, including a second fake Bardiya.

The whole episode is too unbelievable, but figuring out what parts are true is difficult. Most historians today lean toward the idea that Darius and his allies assassinated the real Bardiya son of Cyrus and invented a cover story. However, the fact that Cyrus himself stopped his genealogy at Teispes rather than going all the way up to Achaemenes - who Darius and his heirs presented as an important part of their own royal heritage - leaves scholars scratching their heads. If Cyrus didn't mention Achaemenes, and all of the references to him being an Achaemenid are post-Darius, did Darius make that part up too?

From the way that Darius and Xerxes emphasize their family name, it is clear that being Achaemenid was important in Persia before Darius became king.

And after all that, this where the original question comes back in: if the Achaemenid clan was particularly important in Persia then Cassandane the Achaemenid would be a perfect match for a young prince/king Cyrus the Teispid (as the theoretical independent lineage is known by modern scholars). That would give Cyrus' heirs the legitimacy of being tied to this illustrious "Achaemenes." It would not even be a unique scenario in ancient Persian history. According to Herodotus and Ctesias, Darius married all of Cyrus' living female descendants, and Xerxes was chosen as Darius' heir despite being the third eldest son of the king because Xerxes had the added benefit of being the son of one of those women Atossa, daughter of Cyrus.

The biggest flaw with that theory is thus that we don't have much to verify Herodotus' claim that Cassandane really was an Achaemenid. Herodotus names her father, Pharnaspes, as the father of Otanes, one of Darius co-assassins, as well. However, the Behistun Inscription identifies Otanes as the son of Thukhra. It's entirely possible that Herodotus just has two different Otanes mixed up, several clearly distinct people by that name come up in his Histories. However, it does cast some confusion or doubt on to Pharnaspes in general. Additionally, no source makes any connection between Pharnaspes and the other Achaemenids. We know of a number of Darius' brothers, cousins, and uncles but Pharnaspes is never tied back into the rest of his own family tree, which leaves the open question of where were he and/or his branch of the family through all of this succession crisis.

At the absolute oldest, Pharnaspes should have been a similar age Arsames, Darius' grandfather, but he is not identified with either Ariaramnes or Cyrus I, nor any other names known from that generation of "Achaemenids." It's hard to imagine that in the course of two or three generations, Pharnaspes side of the family just died out with Cassandane and her sons. Of course, if there was an actual Otanes son of Pharnaspes, we're left wondering why he played no role in these events if being an Achaemenid was enough to make Darius king.

With the evidence we have, the debate will remain circular, and ultimately somewhat extraneous. Darius did lie about his legitimacy. Darius does represent a dynastic shift. This debate and theories around are just about figuring out how exactly those things happened.