Delphine is actually a pretty good example of someone in her situation.
Hunted for years, surviving by using everything she's been trained to - spying, infiltration, subterfuge, misdirection, stealth and no doubt - assassination - plus all sorts of other cloak and dagger operations to throw any potential hostiles off her track, but also taking any advantage she can to mess with the plans and actions of known enemies. Something she sees with the Dragonborn - someone fresh into their role, unsure of their power and from her view - easily led. Dealing with the Dragons to her - was a side article, she thought the Thalmor had a hand in their appearance, and If it means messing up Thalmor schemes - she's going to manipulate the Dragonborn as much as she can. It's what she does.
Finding out the Thalmor didn't know about the Dragons was a disappointment - if they had been, then that information would have been a goldmine - seeding that amongst the warring factions and the general population would have led to an uproar against them.
As stated, the freshly empowered Dovahkiin is a very convenient weapon that's literally walked into her tavern and said "So, what's my next task?" and Delphine is pointing them right where she wants the damage done. So as a few have said, no - Delphine does not think highly of the Dragonborn - but she does value the power that is present within them, because that is something she can use. As long as she can keep them focused - which is why she dislikes the Greybeards, a group of powerful men that squander their abilities when it could be used for so much.
Perhaps why she's so insistent on the death of Paarthanax, she's most likely hoping that if she can convince the Dragonborn go through with it, the Greybeards will want nothing more to do with them - and that will get them out of her way to finish "fine tuning" her new weapon into something that can be called upon to act without interference. She has no use for something that she can't use when needed, so if the Dragonborn hesitates/refuses in the killing of Paarthanax - then they can't fulfill the role she wants them to, and is useless from that point on.
So yes, she's a harsh, direct and manipulative harpy of a woman - because that's what her training as a Blade has made her.
Greybeards:
Honestly, I expected more resistance from them over the Paarthanax issue - obviously not happy with it but despite their dedication to him and following of his teachings, they allow you to continue on up that mountain to bring their ancient master his death? When I first picked the option to kill him (a very long time back) I at least expected them to try and blockade me from getting up there. Didn't happen, a few disappointed words and a basic "do what you will" style speech. I'd say they are incredibly dedicated to the meditative and passive nature of the teachings indeed if they are willing to let you carry out the act.
For all intents you kill the Greybeard order along with Paarthanax - the knowledge he has, and the influence he could take to the other Dragons after the defeat of Alduin is immense. No Mortal would bring that kind of status back, and none of the existing Greybeards look like they have long enough left to start teaching new recruits enough to save the order for future generations.
The Blades can survive in secret without the Dragonborn - the Greybeards cannot survive without the Dragon.