Yet another not a mathematician here. Division by zero never converges (as a limit of any generating process, numerical or symbolic) in contrast to division by anything else. There is no amount of zeroes that can add up to a whole number, by definition of zero. Not an infinite amount, just “no” amount.

If you want to handle infinities as numbers, you have to maintain a “progress” somehow, or you end up comparing different stages of processes and get something like 2=1. Much easier to not dig a pit to fall into.