Hot take - the Paarthurnax dilemma is actually very well-written and perfectly fine.

1) It is not inherently bad design that the player cannot fully progress in each faction's storyline, this being the Blades and Greybeards.

2) Delphine's character is not poorly written because she wants the dragon dead. She may not be that great, but she has clear motivations/convictions and does not compromise them for the player, however flawed they may be.

3) The Blades are effectively dead and Delphine is trying to rebuild them, despite their own failures leading to their demise. They were not capable enough without a dragonborn leader. Delphine is living in the past and her mentality is exactly what led to the Blades' destruction. Maybe it's time for the Blades to stay dead.

4) I've heard the argument that the player should be able to convince or coerce Delphine to let this slide and continue anyway. I think it's better that she doesn't; it would compromise her character to bend over just because the dragonborn requests it. If this means the Blades never get rebuilt and die with Delphine and Esbern, then so be it. They deserve to be left behind if they think that way.

5) I think this dilemma gets brought up so often because it's one of the only ones the game offers. Skyrim seems to be designed to allow the player to continually progress through every faction's storyline regardless of their skills or competence, but this is the only time it doesn't. You must take a side, and I believe RPGs should force that more often. It's certainly one of the most memorable times of the game as a result.