Wulfgar won't teach me Unrelenting Force's final word after Arngeir accepts the Horn of Jurgen Windcaller

Along with shutting the gate, also try using Whirlwind Sprint through the gate one more time, shut it and go talk to him.

SOURCE

I'm going to give you the answer you don't want, but with a twist that you do.

Reload from a previous save.

Wait keep reading!

Seriously, reload. But don't do it just once and expect it to work. I encountered the invisible bug from the mage's colledge quest and had to reload a previous save about four times. Save the game just prior to doing the quest, and then reload until it works. Sure it might take a little while but it'll be a heck of a lot less time than restarting from scratch each time. Good luck!