The other reason, which is the one that settles it: if you define it anyway, ordinary arithmetic collapses.
Suppose we declare that one divided by zero equals some value, call it k. Then by the definition of division, k times zero must equal one. But k times zero is zero for every number k. So zero equals one.
And once zero equals one, everything else follows - you can prove any number equals any other, and arithmetic stops distinguishing anything. That is not an inconvenience, it is the whole structure failing.
So the prohibition is not squeamishness. It is that the alternative destroys the system.
Worth knowing that mathematicians have built structures where something like this is allowed - the extended number line, and the projective line where there is a single point at infinity that both directions reach. Those are consistent and useful, particularly in geometry and complex analysis.
What they give up is arithmetic. In those systems certain expressions remain undefined, and the ordinary rules do not all hold. So you can have division by zero, or you can have arithmetic that works, and for everyday purposes the second is worth far more.
That is the honest answer to why nobody explained it: the real reason is that it is a design decision with a cost, not a fact about zero.