First, the same argument above is also an argument that, say, integers, are not "Computer SCIENCE".
More to the point, you might enjoy reading the work of Charles Pierce and other logicians of that era who began to explore many variations on formal logic. Note that just as many operations arise from trinary relations in bivalent logic. Are binary relations "Computer SCIENCE", but not trinary or higher relations? Before you answer, you might want to look into whether all possible relations can be expressed using only binary relations (hint: nope).
Look deeper into the concept of functional completeness (with respect to a subset of operators), which you reference above without naming. You might be able to understand how many of those many trivalent operators are actually necessary to reason with (hint: not very many, hardly more than for bivalent logic, where, as you note, we only tend to use a few, and need not worry about it).
Consider also the relationship between operators folks have identified as useful in bivalent vs trivalent logic (hint: they not picking at random).
Could it be that just as with the 16 binary operators, many of which have relations to one another (e.g. inverses and complements, among others) that the trinary operators could fall into similar groups, which, making the 3^9 number you mentioned seem a whole lot less complex? Could that be why it's neither necessary nor customary to work with all the operators in either sort of logic?
Once you've caught up to state of the art in formal logic as of the 1930's you might have a new perspective -- perhaps you might even begin to let us know when "Computer SCIENCE" will catch up!