The way I see it, the people of Skyrim were not actually all that interested in the dragons. They spent way more time arguing about the whole Empire and Stormcloak War then they did about the dragons or Alduin, and some of the Jarls even gave "kill the dragon" just like they gave "kill this random bandit" quests. I think by this point in their history they are used to nearly world destroying problems just kind of working themselves out, so the dragons took a back seat to the more "widespread" political issues.
When it comes time for the end, the only people that really care are going to be the Greybeards, who are pretty unemotional already, and the Blades, who act like dragonslaying is just something you do, nothing all that special. The rest are only going to hear about you in books and legends, one that will likely sit on some random bandits bookshelf in a hundred years.
Remember Talos, the God that lead to the civil war in the first place? He didn't become a household name everywhere until he literally took over the Empire and conquered the entire continent. We are not there yet.