Any good warrior can "kill" dragons, in the way that their soul is preserved and ready for resurrection from Alduin.
Though that only matters when Alduin is around. If Alduin's gone he can't resurrect them, so any dragon "killed" by a non-Dragonborn would be just as "dead" as one killed by a Dragonborn.
No one knew Alduin was sent forward in time and would come back, since the Tongues that beat him had no idea what the Elder Scroll would do. It was a very desperate move since they didn't really know what it would do and could've backfired horribly. Even Paarthurnax didn't know for sure what happened and needed to stay to keep an eye on the time wound.
Also, it was over 2500 (two thousand five hundred) years between the defeat of Alduin and the arrival of the Dragonguard, who for some reason still believed they needed a Dragonborn to kill dragons, and it was kept that way for another 1000 years or so until Tiber Septim came along and granted safe harbor to the remaining dragons. So that's nearly 40 (forty) generations of men that somehow knew they still needed a Dragonborn to prevent Alduin from resurrecting dragons, when no one knew Alduin was going to come back. That just doesn't make sense to me.
Alduin and his ability to resurrect dragons should've been the stuff of myth and legend well before the Dragonguard showed up. The idea that he'd come back and resurrect fallen dragons should've been stories parents would tell to scare their children by that time. I have a really hard time believing that after a few generations people would continue to believe in earnest Alduin's power and ability to resurrect dragons, let alone a few
dozen generations.
That's why dragons were so rare. Alduin had been sent forward in time, and the ancient Nords killed the rest of the dragons and made the tombs for them that we see around Skyrim.
Except the dragon killing continued through the first era (almost 3000 years) as it was still happening when the Dragonguard came, and through the second (almost 1000 years) since the Reman dynasty hunted dragons, until Tiber Septim came along and offered the remaining ones protection. So for almost 4000 years, dragons that were "killed" stayed "dead" regardless of if they were killed by a Dragonborn or not, yet people continued to believe a Dragonborn was needed to kill dragons? That just doesn't add up.