The main reason that it is quicker to avoid another aircraft by climbing or descending is that we don't consider airspace to be a "true 3D" environment, but rather a series of stacked 2D layers. Separation by 1000' vertically is in most cases considered well clear, but 1000' laterally is considered to be a near miss. This is because aircraft generally maintain a given altitude during most phases of flight, and change altitude much more slowly than they move laterally. So long as you have a fairly precise instrument for determining that vertical separation, you are good to go.
The most modern TCAS systems do indeed "know" whether the best maneuver is to climb or dive to avoid collision, and if both aircraft have the system then they communicate and issue coordinated instructions. If only one aircraft has TCAS, then only that aircraft's crew will receive instructions to climb or descent. If one aircraft has no transponder, or doesn't have altitude reporting, then no avoidance directions can be given, since there is no way to determine the relative positions.
Mike



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