Basically, Delphine and Esbern gave you the middle finger by using "The Blades' Oath" as an excuse. The problem being I don't think that the Blades have ever had any connection with dragons at all
For one, the predecessor of the Blades, the so-called "Dragonguard", did not exist until November 2011. It is said in Skyrim that the Akaviri invaders pledged themselves to Reman Cyrodiil the moment the guy Shouted at them. This does not click with the Pale Pass Fort's ending, in which the remaining Akaviri forces still believe that Reman is a threat. If anything, the Akaviri forces should have reminded one of their headquarters to surrender. Remanada in Oblivion only said that the "loyal knights" descended from old Reman's bodyguards, but who these bodyguards are were never specified
Second, even if the Dragonguard are actually the surrendered Akavir forces, the Blades still had nothing to do with Dragons. The Dragonguard were founded in the 1st Era, and 2 Eras later they're spec ops. You'd think that during the course of these two Eras, in which nothing about "dragon problem" was mentioned, they'd start to use a different oath than "I'll serve you my lord, for you are the Dragonborn"
Third, even if they did not change their oath even by one bit, was there even any oath at all? In Skyrim it is said that they were specifically looking for a Dragonborn, and they were not in Tamriel to look for dragons. Which means that while the Akaviri were probably capable of slaying dragons, dragons were the last thing in their mind. Or at least dragons are second to Dragonborn in their mind
Fourth, let's say that they were looking for dragons "as a side quest" because hey, you don't go out of your home continent that frequently, I doubt that dragons still existed by then. Specifically, that Alduin is still a threat by then.
Remember: there was this man named Ysgramor, who drove the Mer out of Skyrim. You'd think that if dragons was still around rauaging the lands, people would stick together and forget about the war altogether, or at least the Mer would've been fighting off dragons and Ysgramor would've took note about them. Which means that by the time Ysgramor fought off the Mer, either dragons had not yet existed, or dragons were no longer a threat
Fifth, let's say that Alduin appeared after Ysgramor fought off the Mer. The Akaviri forces who would later become the Dragonguard came to Tamriel and was defeated by Reman Cyrodiil. If Alduin was still a threat by then, the Akaviri would be fighting off the dragons and either ally themselves with the Tamrielan forces or just fight them off by themselves. But there is no historical account about dragons near or before Reman Cyrodiil's reign. This means that by the time the Akaviri invaded Tamriel, dragons were no longer a threat. This means that the Akaviri forces, the later-Dragonguard, has nothing to do with Alduin's actions altogether.
Sixth, even if you that Alduin's dragons are the same dragons that the Tsaesci hunted, it is simply not possible. According to the book Mysterious Akavir, the Tsaesci managed to enslave the Red Dragons, and while the Black Dragons fled to Po Tun, a great war was raged that ended up killing all the dragons. Meaning that not a single dragon is left in Akaviri, meaning it is not possible for the dragons to have fled Akaviri to join up with Alduin in Tamriel.
This means that Alduin's dragons and the Akaviri dragons are both different dragons. Perhaps dragons are all the same in Mundus (in the same sense that all Men share the same physical characteristic), but it does not make Alduin's dragons related to Akaviri dragons whatsoever.
This means the Akaviri forces, the later-Dragonguard, couldn't have known just what did Alduin do back in Tamriel (or Skyrim)
Seventh, let's say that perhaps Alduin commanded the dragons in Akavir from the relative safety of Tamriel, perhaps the Akaviri simply had a hatred so deep against dragons in general.
It still does not make 100% sense.
Remember: the Voice was taught to humans during the dragons' reign as a way to fight the dragons, but up until now, aside from being a Dragonborn, only the Nords can use a Shout. Reman Cyrodiil was a Dragonborn, so he does not count (besides, his race was never been specified). Tiber Septim was a Nord, so are the Greybeards, and there is no historical account of the usage of Shout by any other party. This heavily implies that the dragons actually did not enslave the mankind, but simply enslaved the Nords, which would make the whole "The Akavir knows of Alduin's deeds" even more confusing
Lastly, let's say that the Akavir simply hate Dragons
This does not mean that they have anything regarding dragons in their Oath
As I have pointed out before, it is simply not possible for dragons to be a major threat (or perhaps even exist any longer) by the time the Akaviri forces swore loyalty to Reman Cyrodiil. Even if they were, the later-Dragonguard were looking for a Dragonborn to serve him, and not a single time have they ever complained about the lack of dragon-killing during their time as bodyguards and as spec ops
All this boils down into one question:
Which part of Blades' Oath would specify anything about dragons? Especially after such a long absence? And what right do the shadow of Blades have to invoke it?
