What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.
The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division. In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.
