I speak for my personal bug, but many, many others have this on all platforms in varying forms.
I cleared Karthspire and all the puzzles within (they are stupidly simple) a very, very long time ago. I therefore do not have a save to revert to and wouldn't want to anyway. When I got the quest I fast travelled to Sky Haven Temple, Delphine and Esbern arrived and we used the blood seal to open the secret door. I go inside as instructed but Delphine and Esbern both run back to the first puzzle bridge (which is already completed).
I can't continue now, I have tried the following and in cases necessary, a combination:
Resetting AI
Enabling/disabling the characters
Spawning the characters inside the temple (in various locations, including those which trigger the next dialogue)
Force pushing them with shout to entrance
Loading a save before teleporting and killing the dragon and camp outside again with them and then walk through Karthspire
Making them attack me, leading them into the temple, casting calm or hiding
Rotating pillars (they don't rotate, there is no option to now, they are fixed in the wrong order, but bridge is down)
Lowering 2nd bridge
Pressing on the pressure plates
Using console to set the quest to different stages (Partly works, but it really isn't a fix, they are still at the bridge, the temple is dark and I haven't learned first hand the secret or the dialogue)
Fiddling a little with the creation kit, but I really have no idea what I'm doing, I'm not sure what changing any of the parameters do for objects like the bridge or turning pillars..
I've emailed bethesda and they're support team is very arrogant and do not provide me with any help or even if there is a patch in the works to fix this..
I would GREATLY appreciate anyones help on this, I'd even pay, because right now I can't continue the game and it's a complete waste for me, very frustrating.
Cheers