Maybe learn some algebra before you start being pedantic. You can't divide by zero and any notation that does is shorthand for something else. There is no way to define 6/0 in the same way as 6/3 and the fact that you need a whole-ass metric space so you can have convergent sequences shows that.

And n^0^ is not just arbitrary but the definition of an empty sequence of an operation is that it yields the neutral element. The empty sum yields 0, the empty product yields 1 and n^0^, in the most naive and direct interpretation, Is n multiplied by itself no times, hence an empty product and therefore yields 1 (there is a caveat for 0^0^ but that's like a whole lecture). Sure that is arbitrary in the sense that any axiomatic system is arbitrary but not in the /0 sense where it keeps changing based on context and no self-consistent axiomatic system is possible where /0 is assigned a value.