Not a very satisfying answer (feels like if a satisfying answer to this question was possible, maths would look very different) but I think the trick to not getting annoyed by it is to shift your perspective rather than try and redefine maths. It's not that division is a badly behaved operator; it's that division isn't an operator. It's a shorthand for multiplication by the reciprocal. Since the reciprocal of 0 doesn't exist, we can no more multiply by it than we can compute 5 + the sound of one hand clapping. And we don't fret over how the addition operator's inadequacy there is unsatisfying!
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Not a very satisfying answer (feels like if a satisfying answer to this question was possible, maths would look very different) but I think the trick to not getting annoyed by it is to shift your perspective rather than try and redefine maths. It's not that division is a badly behaved operator; it's that division isn't an operator. It's a shorthand for multiplication by the reciprocal. Since the reciprocal of 0 doesn't exist, we can no more multiply by it than we can compute 5 + the sound of one hand clapping. And we don't fret over how the addition operator's inadequacy there is unsatisfying!