An Empire (or anyone else) with the Divine Right to Rule and the endorsement of the gods is always going to be more legitimate than other forms of government (at least to its subjects) however whilst it may make them less legitimate not to have it, I would say there is no "loss of legitimacy" in it's entirety.
There are other forms of legitimacy, such as the consent and endorsement of those that you rule. The Septim dynasty was no doubt in a far better position to claim Right to Rule due to (claiming they had) endorsement from the gods and its people, but they started out in a way which beings me to another kind of legitimacy. In the same way Titus Mede seized control of the Empire, Tiber Septim took control of the Empire and Tamriel by Right of Conquest. Some may even argue that the methods Tiber Septim used would detract from his legitimacy, but I digress. If you can take it and hold it, it's yours. If you can't hold it, you had no business being there. In a way, you can apply this train of thought to the Septims around the time of the Oblivion Crisis, although I doubt that opinion is popular. In Chim-el Adabal: A Ballad, it is actually stated that Akatosh gave Alessia the Amulet of Kings as her reward for conquering Cyrod.
Heavy leaning on the whole concept of Divine Right to Rule led to some of the worst Emperors (Septim, at that) that Tamriel has ever had, and was arguably incredibly detrimental. Which brings us to Right by Qualification. Concepts of lineage (e.g. eldest son takes the throne) and disagreements about lineage led to civil wars and uprisings, so much so that the Empire seldom bothered to appoint a leader because they could actually lead. This is the system that the empire follows as a monarchical (Right by Birth) and despotic arrangement, which I have no problems with, so Right by Qualification is difficult to achieve in these forms of government, however sometimes they get lucky and actually choose a competent ruler. Regardless, it is relevant to concepts of rights to rulership.
Divine Right of Kings isn't everything, and a terrible Septim ruler can be less legitimate than a highly successful non-Septim one, although having direct contact and verbal agreement with the gods does help your case. Akatosh made the agreement with Alessia because they took pity on the plight of Men, which is why the Covenant was created. The gods may no longer feel, after centuries of freedom from enslavement and development, that it is no longer necessary, and the ending of the Oblivion Crisis pretty much supports that idea. Whilst mortals believe that being Dragonborn is a hereditary connection that bestows Right to Rulership (thanks to more than one Dragonborn dynasty) nobody knows why Akatosh bestows the Dragonblood. We actually don't know for sure if being Dragonborn is hereditary or solely at the discretion of Akatosh, although I'd argue that it's the latter. Despite this, a mortal lineage claiming it has right to rule because it has an unrelated agreement with the gods is a mortal machination, not actual endorsement from the gods.
It's now more accurate to say that without the necessity of the Amulet of Kings and the desire to seal Oblivion off from Mundus (and the Covenant), any old Dragonborn (or even Septim) Emperor now has about the same legitimacy as an ordinary person, because the Covenant has been concluded, the only people who would say otherwise are those mortals who associate being Dragonborn with Right to Rule, which was a political idea brought about solely by the Covenant that Alessia and Akatosh started. To say Akatosh bestows the Dragonblood on those he deems worthy to rule is incorrect. In the end, one of Alessia's Dragonborn descendents could have fulfilled the Covenant without ever ruling, and people perceiving her ruling descendants as more worthy to the Divine Right of Kings is actually a subjective view.
Conclusively, "Rights to Rulership" are judged by these standards in the real world and in Tamriel (just with different gods and less concrete claims to Divine Right of Kings), but whether rulers in question fit (or fitted) the criteria is subjective.
