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"I find it very difficult to figure how putting a mass into the trajectory of the threat asteroid, such that the threat is attracted to that mass enough to be pulled out of the threat trajectory, can possibly be more efficient than simply colliding with the threat asteroid."

The idea of crashing a spacecraft into an asteroid to alter its trajectory is seductive because, at first thought, it appears to be a simple, predictable game of celestial billiards. In other words, if we hit the threat asteroid at point X with mass Y traveling at velocity Z then A, B, and C will result. They script variations of this all the time in the killer asteroid movies Hollywood produces, so it must be possible, right? It’s accomplished in a flash and everybody can breathe a sigh of relief and go on with their lives. Who wants to fret over a boring gravity tug on a killer asteroid that drags on for 20 years?

Unfortunately, the reality of the situation is very different. A solution in the collision scenario can be calculated with barely an acceptable degree of certainty only if we know beforehand the exact material composition of the threat asteroid in all regions and its mass distribution to a degree of certainty approaching what we know about a billiard ball. To obtain that scope and depth of knowledge about a small celestial object hurtling through space many millions of miles from Earth is essentially hopeless. Even if we had the requisite knowledge of the asteroid, the solution might show a sufficient mass for a collision scenario couldn’t be constructed and delivered from Earth in time.

A basic rule of killer asteroid wrangling has something in common with the ethical practice of medicine described in the Hippocratic Oath, i.e., “First, do no harm”. Since the precise consequences of a collision can’t be predicted or quantified to an acceptable degree of certainty, it’s wise to attempt to alter the trajectory of the asteroid without adding the risk of making the problem worse; especially if the result might be multiple asteroid fragments. The stakes would be too high to do otherwise if an approaching asteroid threatened Earth with an extinction impact event.

The idea of crashing a spacecraft into an asteroid to alter its trajectory is seductive because, at first thought, it appears to be a simple, predictable game of celestial billiards. In other words, if we hit the threat asteroid at point X with mass Y traveling at velocity Z then A, B, and C will result. They script variations of this all the time in the killer asteroid movies Hollywood produces, so it must be possible, right? It’s accomplished in a flash and everybody can breathe a sigh of relief and go on with their lives. Who wants to fret over a boring gravity tug on a killer asteroid that drags on for 20 years?

Unfortunately, the reality of the situation is very different. A solution in the collision scenario can be calculated with barely an acceptable degree of certainty only if we know beforehand the exact material composition of the threat asteroid in all regions and its mass distribution to a degree of certainty approaching what we know about a billiard ball.

I confess that in advocating direct impact I was taking for granted that the collision would be perfectly elastic and that the target would stay in one piece. And it seems to me that, unless the target is actually an accretion of dust, those assumptions should be sound. They could be made more so by detonating explosives on board the rocket to shred it into small fragments a moment before impact.
It is of course true that as the mass of a distant asteroid would not be known a priori, the extent of the effect of a given impulse on the velocity of the target would also not be known a priori.

Thank your for your erudite explanation of the problem.