I haven't dug into Blades lore, but what I've gathered makes a certain amount of sense and is kind of cool: The Blades were once an order of dragon-slaying warriors, traditionally led by the greatest dragon-slayers of all: the Dragonborn. Once the dragons were (apparently) extinct, the Blades continued to serve the Dragonborn Septim dynasty, evolving over time into an elite royal guard of sorts. I have no trouble accepting that historical sequence of events.
And since the the death of Martin -- and the end of the Septim bloodline -- the Blades have slipped into obscurity with a handful of individuals trying to preserve their traditions. Again, I have no trouble accepting this. Makes perfect sense.
So now there's a new dragon threat and a new Dragonborn. Suddenly Dephine's obscure order has a reason to exist again! She has something to rally behind! Of course she's going to be all over that. But while the Dragonborn has the natural talent she needs to accomplish her mission, she and Esbern -- not the Dragonborn -- have the knowledge required to make it work. So of course she's going to take a leadership role, directing the Dragonborn as needed to end the dragon threat as quickly and efficiently as possible. Once again, I have no trouble accepting this. Makes perfect sense.
If someone more well-versed in TES lore comes along and tells me why I'm wrong, I may concede. But as it stands, I have no problem with Blades lore or Delphine's actions.
I have dug deep into the Lore pages of UESP, and judging from http://www.uesp.net/wiki/Lore:Blades http://www.uesp.net/wiki/Lore:Tiber_Septim http://www.uesp.net/wiki/Lore:Dragons http://www.uesp.net/wiki/Lore:Alduin http://www.uesp.net/wiki/Lore:Akaviri the Blades were formed in the Second Era by Tiber Septim, based off of the Dragonguard of Reman in the FIRST Era
HOWEVER, the height of dragonic power in Tamriel was ended during the Merethic Era, during the Dragon War. By the time of the 1E, most Dragons had been wiped out, with only a small number living in isolation among mountains. It is true that the Akaviri, and later the Dragonguard slayed Dragons.
But by the time of Tiber Septim, 500 years later, Dragons had become so rare that they were already becoming the stuff of legends. Tiber Septim offered safety to the very few willing to swear fealty to him, but eventually they died off and truely did become legends, with the only people who claimed to know of existing Dragons being the Akatosh Chantry (I can not find any proof supporting thier claim of having 'tame' dragons).
There are rumors of dragons during Third Era, but no proof to support those claims. The Blades were never dragon slayers (because Tiber Septim, despite being a Dragonborn, was not a dragon slayer), although their predecessors (the Dragonguard) were. The Blades have nothing to do with Dragons, only with the Emperors (who happened to be Dragonborn).
Based on this information, it is my educated opinion that Delphine and Esbern are NOT trying to rebuild the Blades, but rather trying to rebuild the Dragonguard. In which case their desire to kill every single dragon in existance is completely understandable.
From this point foward, I will refer to the 'new' Blades as what they REALLY are, the Dragonguard.