Of course! That's the easy part. But if we nudge the asteroid a small bit, it might have an effect on an asteroid it passes within within several hundred miles of a few years from now. Then that asteroid (#2) now has a slightly different orbit/movement around the sun/Jupiter/asteroid #3. Now, asteroid #3's movements are now modified so that in 50 years, it collides with asteroid #4. This results in asteroid #4 turning into asteroids #5-15. And one of othose new asteroids ends up having a trajectory that intersects with Earth's orbit 75 years from now.
Far fetched? Sure! But are you telling me that the mission planners looked at every single one of those possibilities and concluded that wouldn't happen? I doubt it. For one thing, they don't have the time or money. But more importantly, they can't calculate a domino effect when they aren't even sure how many dominoes there are, let alone how big they are, what their mass is and what their orbits are.
My whole point isn't that the above will happen. My whole point is that the above is possible under the laws of orbital mechanics as we understand them because we don't fully understand 100% what will happen when we hit that asteroid and what objects are floating in and around the asteroid we hit.
The asteroid belt is sort of a rock soup. Bits and pieces hit each other with some frequency, altering their trajectories in ways that are, in aggregate, beyond our ability to model.
I will grant you the hyper-theoretical infinitesimal chance that an experiment like DART could lead to a bad consequence very far in the future.
BUT it's just as likely that some unplanned effect of pebble tapping rock tapping boulder would lead to the DIVERSION of an asteroid that would have hit us many years from now. This cancels out the tiny unpredictable risks you are hyping. You missed this factor in your logic.
This brings us back to: we are not safe from asteroids. This experiment does not directly measurably increase or decrease risk of us getting hit. It does give us data that can lead to tools we can use against a one-day-identified threat. In this way, the effort is a net positive for us.