The blades in this game dont actually realize what thier background or oatsh actually mean. delphie has been so far away from the group that she is basing all her assumptions on how the blades were by the Dragonguards. I REALLY dispised thier ultimatum. They are to serve and protect the dragonborn after they gave thier oath to Tiber after he whupped thier armys ass. And Tiber Septim gave his oath of protection to the dragons that helped him or stayed outta his way during his conquest, that is why pretty much every emperor since Tiber have forbade the Blades of going after Paar because they were HONORING Tiber Septim's oath to the dragons that have helped/stayed outta the way...and Paar has stayed outta the way PLUS Tiber knew that Paar was the leader of the Greybeards when they annouced him being the dragonborn. Its problemly because of Paar that Tiber septim realized that dragosn werent just mindless beasts ands they could be reasoned with.
It was Reman Cyrodiil, not Tiber Septim, who got the Dragonguards. Tiber Septim was born after the Tsaesci rule, the 2nd Era. Tiber Septim ushered in the 3rd Era
The Akaveri were Dragonslayers before they became an army of Dragonslayers that served the Dragonborn. Their first and most primary purpose was to be Dragonslayers. Delphine tells you right at the start that the Blades have been searching for their purpose since the last Dragonborn more than 200 years ago (Oblivion-Martin was his name.)
In the game there's a book that walks you through how the Blades were in all the capital cities of Tamriel and that as the Emperor's personal guard they were also keepers of intelligence about most groups in Tamriel and about certain individuals... like the Thalmor for the Aldmeri Dominion. In fact, I would suggest that it's exactly this power that marked the Blades for complete extermination by the Thalmor when more than 100 of them were found and killed inside the borders of the Aldmeri Dominion and then the Blades were ordered to be hunted and killed everywhere else.
Delphine has a Dragonborn to serve, swore to be a Blade and protect the Emperor and then was hunted for her devotion. Now Dragons are back and her primary purpose is clear again. No matter how she's going to go about it, she is a Dragonslayer and she serves the Dragonborn. Except we "fall in love" with Paarthurnax and spoil her perfect little future. A Dragonborn who is not a Dragonslayer wasn't in Delphine's plans because it makes no sense. We're born to absorb their souls and keep them from coming back over and over again, why wouldn't we be driven to destroy them all?
In her mind you need to be educated and so she's going to "educate" you by making demands. She's entirely off her nut, and besides that, she's being insubordinate. We don't have any option that allows us that answer and so we abandon the temple that can only be opened by us and we leave her and Esbern in our temple to plot and plan and become a future problem because we can't take control over them.
Her motives make perfect sense. She's a Dragonslayer who serves the greatest Dragonslayer of all time, the Dragonborn. We're the ones not making sense... if you look at it that way. I don't have to agree at all to understand it.
They've been looking for a Dragonborn to serve. 2 eras ago Akaviri Army came to Cyrodiil only to beg forgiveness to Reman after he Shouted them. Perhaps he Shouted so loud their eardrums broke, but they surrendered the moment they heard his Shout. Had Reman tried to use his Shout a bit earlier, the war would've been shorter
I think I read somewhere in an in-game book that they spouted Akaviri equivalents of "we didn't mean to" and "we were looking for you" to Reman.
They have never been dragonslayers. They were ronins looking for a lord to serve to upgrade into samurai, and after a few centuries, they upgraded into ninjas. Now after some more centuries, they downgraded into 2 crazy citizens. They COULD kill dragons, yes, but so are any other Akaviri. It's nothing special.
Perhaps it's special because dragons are awesome by default, but so are Daedras, and we have Vigilants of Stendarr to compare (which is quite a crappy comparison, because both are crappy at the moment)