IMHO I think the problem here is that most people focus on the "wrong side" of 1/0.

I mean, what the mathematical definition of division says is not that 1/0 is indeed something and that that something is "undefined" or "NaN" or anything else really. What it says is "I cannot do 1/0, the division operation a/b does not apply when b is 0".

So 1/0 is not a thing in itself in mathematics; it's something which cannot be operated.

Now, back some time ago, "division by zero" simply threw an error. It signaled "this is not something that can be done". undefined, NaN, anything else, including 0, is not really something that has a mathematical justification. It's merely a practical approach to encapsulate that error into some form of pseudo-value to control it to some extent.

Personally, I don't really see how 1/0 = 0 is better than 1/0 = NaN or "undefined" or "Infinity".