Your last paragraph is correct, but only very close to the collision. Things in orbit around the sun don't move in straight lines, even if their paths are going to intersect. The earlier you see it, the less it's moving straight at you.
(Of course, if it's going to hit you, the faster it's going, the more straight at you its path is at the same distance. But for the same amount of "not straight at you", faster leaves a bigger streak.)
Yeah given that were talking about objects that are colliding with the earth, the faster they will come closer to us the less time we'll have to spot them
(Of course, if it's going to hit you, the faster it's going, the more straight at you its path is at the same distance. But for the same amount of "not straight at you", faster leaves a bigger streak.)