Sorry, but probably not. The primary reason this won’t work is that invariably the object is tumbling or rotating. So there will be no place to place the rocket motor that reliably steers the asteroid away from the earth. Similarly, the tumbling prevents tethering it and dragging it away.
I don’t have the cite right now, but I remember hearing about a (NASA?) paper addressing the “How to Deflect an Asteroid” issue and their answer was quite surprising. Given enough time before impact, simply construct a large, heavy spacecraft and park that spacecraft close to the asteroid, in the direction that you wish the asteroid to move. The gravitational attraction of the spacecraft will gradually shift the path of the asteroid.
Of course, if you park the spacecraft and then do nothing, the asteroid and spacecraft will mutually attract each other and (I assume) collide. I believe the spacecraft has to use thrusters to maintain its distance from the asteroid. Over time, the asteroid is gravitationally drawn to the spacecraft, and as the spacecraft maintains its distance, the subtle alterations add up – hopefully enough to miss the earth.
Obviously this method requires a long time to work. It’s useless for an asteroid weeks away from impact.
I’d say that the most dangerous aspect of the Oort Cloud being disturbed is that comets in the Oort Cloud that end up entering the inner solar system would come from all directions in the sky with relatively equal frequency rather than coming from a direction close to the plane in which the large planets including the Earth revolve (roughly coplanar with the plane through Earth’s celestial equator) about the sun.
Comet Hale Bopp is an example of a comet that came from far above Earth’s celestial equator. It was discovered at a distance of about 7.2 AU by amateur astronomers and comet hunters, for which the comet was named. The big problem is that comets are not very bright and usually invisible to the naked eye at this distance, and much less bright than Hale-Bopp was (it was an extremely large comet), and are usually moving much faster than asteroids are. We can catalogue close to 90+% of near earth asteroids, and it is unlikely that their orbits will change or that one will come from somewhere completely unexpected. But with long period comets, they can come from anywhere, and it might even have been the first time that they’ve ever been through the inner solar system.
If it’s rotating, it’s doing so about an axis. So you can still push it through the axis of rotation with no problem. Even if the axis of rotation is perfectly in line with the collision course, you can put a rocket away from the axis and just fire it when it’s in the proper position so that on each rotation you push it off course.
Assuming we can get to the asteroid, dealing with the tumble is not a particularly hard engineering problem to solve.
The angular momentum of the asteroid is a constant vector, but that doesn’t mean that its rotation axis is fixed relative to the asteroid. The key phrase here is “torque-free precession”.
Still, if the “rocket motor” were something like an ion thruster, it could presumably be turned on and off. If it were a chemical rocket, it could be used up in a short period of time, so it would just have to be timed to go off at the right moment.
Ok, I went and looked up the references to what I had heard. Here is a Wiki article about all of the ways to deflect an asteroid:
Here is the section of that article that talks about the “Gravitational Tractor” method that I mentioned in my write up (without knowing the actual name of the effect):
And here is the full Wiki article on the Gravitational Tractor.
PROVIDED there is enough time for the Gravitational Tractor to work, it seems to be the best method. There is no risk of breaking up the asteroid. It works as well on densely packed material as on “rubble pile” asteroids.