She could have gotten the tablet herself and headed off on her own. Why wait for you? Like I said she didn't even know who you are or if you were even capable of killing a dragon? Her whole story is full of holes.
Sorry but you're inventing these holes.
There is no reason for her to get the stone tablet so long as someone gets it. The Blades haven't been dragon slayers for over a thousand years. They've been primarily spies up until that point. And spies(realistic spies) get other people to do the work for them whenever possible rather then risk themselves. (on that point she could have hired the mages you find inside to retrieve the horn and place the letter.). Plus she lives in the shadow of the burrow she'd risk her cover if one of the townsfolk saw her heading up there and coming back. More importantly she can't translate the stone on her own so she needs Farengar for that and he's the one who sends you after the stone.
And dozens of people even powerful people send you on errands they are perfectly capable of doing themselves why should Delphine be any different.
While she's waiting for Farengar to translate the tablet and transcribe the map of the burial sites your killing a dragon.
How did she know you killing the dragon at the tower would show you were dragonborn?
She didn't this is something you're inventing. She had absolutely nothing to do with you going to fight that dragon. We have no way of knowing what she was planning before the Greybeards summoned you. But everyone in Skyrim heard the Greybeards summon you so that would change her course of action. Instead of investigating the burrial sites she decides it best to try and recruit the guy who may or may not possess the power to eat dragon souls. As Delphine said "The Greybeards are nothing if not predictable." so she headed off to steal the horn. Either personally or hiring the mage you find fighting inside.