Why would we assume Cochrane didn't have an inertia-damping doodad?
And space is full of shrapnel today. Nobody can get at it, and nobody wants to. But eventually somebody will have to. And so, whatever Worf's blaster left behind would just be swept away by whatever futuro-brooms Earth uses in the late 21st century for keeping its orbits clear.
On the other hand, Earth in Trek is full of alien junk and time travel artifacts such as Braxton's timeship wreck. No doubt these will be identified as such after Earth contacts aliens who have can openers just like that thing found in Louisiana or catches a time traveler flying a pod whose centurial brakespool matches the thing everybody thought was a Quin dynasty wash basin. But not before. And what's the big deal about finding out that aliens exist after 2063?
Timo Saloniemi