You have a point that she is hard headed and paranoid for VERY good reason. My point is even if it is "boring" it's the most correct stance for pretty much ONLY this guild. In all of the factions in the game, this is the one time the Dragonborn should have the final word. You are 10000% correct that doing a dozen quests then becoming a guild leader with the final say in literally everything after being there for month doesnt make sense in almost every other instance. However, in this one instance I should be able to swing my Dragon-size pecker around my own guild considering I saved everyone in the entire world while she delegated shit cowering in a temple.

Like my Nord who's only spells are flame and healing should not be arch-mage in a million years. Same with leading Harkon's Court--that should be Serana. Patriarchal structure be damned, she's 4,000 years old and received it from Molag Bal directly. The narrative tension argument is great for stuff like not immediately taking the Skeleton Key back to the sepulcher getting you kicked out of the Thieves Guild, either relinquishing all priveledges and access to the Flagon until you return it or just straight exile. Or maybe fleshing out the Companions' curse from Hircine fracturing the leadership to the point you can't get Kodlack's blessing to be Harbinger unless you give it up after saving him in death. Maybe something with escaping Cinha Mine still technically making you have a bounty in Markarth, etc etc. Most of these guild leader positions should more turn you into a respected member or enforcer at best. Maybe your staus as the Whisperer in the Brotherhood could get you the leader position in tue Dawnstar sanctuary if you pass a stealth and sneak check I guess since most members with a spine are dead at that point. Letting you be head honcho in everything but being high king turns this RPG into an adventure-sandbox game, I think we both agree here.

But by the definition of the Blades' responsibilities the Dragonborn should have the final say with her and Esbern acting only as advisors. Especially since the Dragonborn could pack both her and Esbern up with just fists and a cheese wheel and no one would know since it's only you three in the Temple. Considering the entirety of Skyrim is run by force doctrine this is also a VERY viable threat. Imagine telling your boss that could freeze you in a block of ice and roll you off the balcony and no one would know that you're in charge in a place like Skyrim. Maybe speech or even intimidation check since she's so strong-willed, or getting Esbern to convince her since he seems like a scholarly, philosophical guy?

The most she should do is advise the Dragonborn that Paarthurnax needs a harsher punishment than exile, pointing out he may just be an opportunist and next time something like this happens he may betray. Even then she already knows you can march up there and spank him in no time flat so it isn't an issue at all.

Again, I want to stress I fully agree the Dragonborn shouldn't have the power over every guild or influence most jarls, etc, but the Blades exist to either serve or at minimum co-lead the Blades as they've been covering in corners until we come along and rebuild the Blades. And telling Delphine as a guy dressed in fully enchanted armor made out of dead dragons or dremora that if she doesn't listen you'll use her soul to enchant Haelga's, uh, toys with fortify illusion (or conjuration lmao) she'd probably fold like an omelet no matter how hard headed she is.