My one and only reason for disliking Delphine is that she has a tendency to treat the Last Dragonborn like a weapon / attack dog rather than a person. She throws them at situations without even giving them proper training.
If we go by the main quest alone, Delphine has the Dragonborn fight Sahloknir almost completely by themselves after only just having learned Unrelenting Force and Whirlwind Sprint. Even for someone who is technically a demigod, that is a highly dangerous situation to be put in. Yes, they have defeated Mirmulnir before this, but then they had a dozen Whiterun guardsmen to help them.
Delphine then goes on to pull similar tricks by having the Dragonborn go into the belly of beast itself by infiltrating the Thalmor Embassy regardless of whether the Dragonborn has any experience in the art of stealth or not.
Then, as the Dragonborn is fully focused on preparing for their fight with Alduin, Delphine approaches them and demands they kill Paarthurnax. I understand why Delphine would want him dead and I understand that this quest is about asking oneself whether retribution is still something that should be enacted upon someone who is seemingly doing his best to atone for his past crimes, but you can see how this just puts more pressure on the Dragonborn during the most stressful time in their life, no?
I don't care about Delphine being harsh and mean - she can be as mean as she wants, she's a literal war veteran - and I don't care about her not borderline woshiping the Dragonborn either. Hell, I don't even care that she goes against the popular idea of letting Paarthurnax live. If anything, it feels refreshing to have a character who isn't afraid to let her true opinions be known. She's an interesting character like that!
The simple fact of the matter is simply that, in being a complex character, Delphine is both right and wrong at once. She generally has good intentions but she operates under the philosophy of the ends justifying the means to get there, which is reflected in how she throws the Dragonborn at every major issue she has throughout the game. I'm also not a fan at all of her hardline pro-retribution stance in the Paarthurnax dilemma. It just rubs me the wrong way, especially when I consider that I've written my own Dragonborn as a character who struggles with seeing himself as a good person; he does his utmost best to do the most good for the most amount of people, only for there to be this one person who implicitly communicates to him that he'll never be "good enough" because he disagrees with her once and is frustrated by how she treats him.
I hope I made any sort of sense here. There's a lot of interesting discussion surrounding Delphine, but I don't fall neatly into one of the two major camps of "I hate Delphine and anyone who likes her is stupid" and "I like Delphine and anyone who dislikes her is stupid" so I wanted to add my own opinion to the pile here.
In summary: I like Delphine from a meta perspective because she provides an interesting counterweight to Paarthurnax and the Greybeards, but I don't like her from the perspective of looking at how she treats the Dragonborn.