that should be an option. Since the Blades are incapable of recruiting without your help, and both Delphine and Esbern are old and unable to have kids... I say the Blades are dead. Good thing they got all that entitled attitude. Delphine laying out the "kill him or else" seals the Blade's fate.
I personally believe that whether or not you helped them rebuild the Blades, the organization will continue. There is no way that Delphine and Esbern are the sole remaining Blades, since they were stationed all across Tamriel during the Great War. If two survived, others must have as well. Having said that, I always choose to help the Blades. I usually finish the CW before I get to Season Unending, that way I don't have to deal with killing Paarthurnax, but either way, I stay loyal to them. The Greybeards are a peaceful group, but they are too peaceful. They are willing to sit back and relax while the world ends - "Maybe this is the time the world is meant to end" - I hate Arengir.
I go around Skyrim trying to obliterate any cults that I find. That includes Daedric as well as Aedric. The Greybeards are Kynareth's cult. I cannot kill them, but I have no intention of serving their ways. Their way of life has made them useless and at least the Blades want to do SOMETHING to stop the madness. Extreme, yes. Hypocritical, yes. But in times of peril, people have to do what they don't want to. If it is between the Greybeards letting the world end, or the Blades committing genocide, I will stay with the Blades. At least genocide of an ancient and long-ago dead race is better than xenocide against every species of men and mer.