Now Paarthurnax explains this to you, that Alduin, when he was defeated the last time just got pushed from that time to the future. That future has arrived. The three Nordic Heroes ripped the fabric of time using the Elder Scroll to move him out of their time when they could not defeat him, essentially putting off doomsday.
So now you've got Alduin bringing about the TES version of Ragnorak, and the Thalmor not knowing anything about why the dragons returned, yet they still want to unmake humans from the world and return the world to Mer only, not realizing that Alduin will be their undoing as well. And they don't care. They are only after power to transcend the mundane and destroy mankind.
But the Greybeards don't understand the Thalmor. So they opt for the truce or stalemate which plays into the Thalmor hands, thus trading one crisis for another.

On my latest playthrough, I plan on siding with the beards as much as possible. I don't think the player can truly be a greybeard type, since they're sent to the world as an active agent of akatosh. Not to be a monk. That said, there's wisdom to be had from their position. And Ulfric's and the Empire's as well. With Ulfric, I recognize his need to "do something". To try. From the empire, I learn that uniting people is a good thing..their mistake is selling themselves out too much for the sake of peace. From the greybeards, I learn that I must transcend petty squables as much as possible. The other thing I like about being neutral is that I'm not required to take oaths for anyone. For me personally, I dislike being obliged to any one person like that. I would never do it in real life. I'm not big on patriotism either. I was always the kid who sat down during national anthems, and piss my militaristic dad off. Heh. It just didn't make sense to me. So I'd like to play a character like that now. It's a bit closer to myself.

edit: Actually, I might say my position is closer to a Psjic than a Greybeard. They're neutral too, but they understand the Thalmor threat.