No. That is specific to the Blades. Delphine herself says very early on that the Blades exist to serve the Dragonborn, and that they are sworn to guard them, that it is their very purpose that they have been searching for, for the last 200 years.
As the Dragonborn, we not only stop Alduin, but we reunited the last two Blades, who would have likely perished or faded into obscurity without us, and re-opened their temple using our own blood.
If they are willing to turn on the player after all that, they clearly were never loyal to us to begin with.
Doesn't Arngeir comment that one of the reasons he dislikes the Blades are because while they claim to serve the Dragonborn, in reality they try to control the Dragonborn?
Anyways, they really are an irritating pair. When Delphine said that unless I killed Paarthanax, the Blades weren't going to help me again, I'd love an option to slap Delphine and tell her: "You are sworn to serve the Dragonborn, Delphine. You and all the Blades. You'd still be hiding in your backwater inn from the Thalmor if I hadn't jumped through all the hoops you put me through. You wouldn't have this secluded temple, hidden from the rest of the world and the Thalmor to rebuild the Blades. The Blades would still be limited to you and Esbern if I hadn't brought willing recruits to you. If not for me, when you and Esbern died, the Blades would become nothing more than a footnote in history, forgotten to all but scholars; perhaps a deserved fate, given how badly you all failed to protect the Empire. So do not think to order me around. I am the Dragonborn, and you are sworn to serve me. I am not your lapdog, despite what you may have convinced yourself".
I actually took my weapon out and smashed Delphines face in and sent her sprawling onto the floor once she delievered that ultimatum to me. I then reloaded an earlier save.