They let Tiber fly all those dragons, over Cyrodiil and Hammerfell. Just saying...
They must all agree on favorite dragonborns, so they let them get away with anything.
Or it could be that the Blades alive at the time were smarter, either by giving their oaths allowance for Dragons that had themselves sworn to serve the Dragonborn or by putting them on a hit list for another day.
I think Delphine's just kinda pissed and bitter and wants to take it out on an enemy. I mean the Thalmor basically ruined her life as a Blade, now she's a Dominion fugitive. Seeing as how the Dragons were the old enemies of the Blades when they were Reman's Dragonguard, she just wants a way to vent out the anger and pain.
You notice, too, that she is very eager to have it all be the Thalmor's fault. It would have been so clean for her after thirty years of running to basically have one of her oaths as a Blade, even an oath that was technically redundant since the early 3rd Era, involve killing Thalmor. Then it winds up that the Thalmor are just as confused as anyone else.
And, of course, the Blades were already desperate for a purpose back when Martin transformed. With no Dragonborn to protect and no Dragons to kill, you have a bunch of people with too much dedication and nothing to be dedicated to. When the Dragons vanished, their purpose switched to becoming the personal eyes-and-ears to the Emperor. When you have neither, then you'll jump at the first that resurfaces. If the second comes back at the same time, well then it's all up to the individual Blades. The two you find in Skyrim are a crazy old man with a justifiable Alduin obsession and prophetic dreams and a bitter middle-aged woman who may not be the most sane.