Tiber Septim didn't kill all the dragons. He even let one serve in his army. He was also allied with the Greybeards who were led by Paarthurnax at the time (though whether this was known, I can't say, but the Greybeards didn't particularly keep it a secret once they knew you were Dragonborn).
And the Blades had left Dragon-Slaying behind them by then - given that they were almost all dead - so what's your point? Dragons are tearing [censored] up now so that remaining Blades have gone back to their roots - given that they no longer officially exist and can't exactly guard the Emperor - Paarthurnax has been identified as a possible threat and not without good reason.
Given that almost everyone on here has been like, "The Blades svck because Paarthurnax didn't seem like a bad-guy", testing the loyalty of their supposed leader seems to have been a move in the right direction. The Blades - and the T'saesci before them - believed that a Dragonborn made the best Dragon-
Slayer, when Dragons are roaming freely and tearing [censored] up they want to follow someone who will kill Dragons, not let personal feelings get in the way of what they consider justice. That is probably why they say they won't trust you till you kill Paarthurnax.
EDIT: I've just came to a thought. On all of these threads people have been moaning about the Blades wanting you to be their pawn in killing Paarthurnax. What you all also seem to want is for the Blades to serve you unquestioningly because you are the Dragonborn. Thee Blades follow a Dragonborn because that Dragonborn is the ultimate Dragonslayer, if he doesn't want to do that but save the Dragons then how can you expect them to still follow your orders. Except you still seem to want them to because they are sworn to follow the Dragonborn apparently, who's using who as a pawn here? You, for wanting the Blades unquestioning loyalty even when doing the complete opposite of what their order stands for.