I've never been able to bring myself to take "The Blades" seriously in this game. Delphine is a straight out lunatic and never shows the Dragonborn any kind of respect. The Greybeards are wary due to their adherence to The Way of the Voice. They're wary, but they also don't seek to manipulate your path. Delphine treats you like a fresh faced recruit from the start and gives you marching orders without bothering to even ask if you care to join this particular fight. She shows up to the Peace Accord in full regalia, as if they're a legitimate player in the game of politics.
When House Telvanni has a bigger presence, both in-game and according to the narrative, then your organization, you really shouldn't be trying to act like you're more then you are.
If it had been an option, I would have told the Thalmor right where to look to ferret them out. As it is, I interact with them as the narrative demands, and I leave them to their delusions in a crumbling ruin. When given the choice between choosing between a centuries old creature that has lived in seclusion and preaches meditation and wise handling of the language of pure magic vs a clearly unhinged woman whose war against the elves lead to a wagon cart full of heads and who wants to seize upon an ancient heritage she has zero training for and even less ability towards...
I'm going to side with the Dragon on this one.