By this line of reasoning, isn’t 7/7 also zero?

If you’re using the word “add” to talk about division you’re thinking about it wrong. Division is the inverse of multiplication, it has nothing to do with addition.

I’ll give my standard answer for questions about 0/0:

Division, as the inverse of multiplication, can be thought of as a question. That question is “what do I need to multiply this by to get that?”. For example, when you say “8/2”, you are asking what number you need to multiply 2 by to get 8, the answer of course being 4.

With 0/0, you are asking the question “what number do I need to multiply 0 by to get 0”? 0 would work. But so would 1, 2, π, any other number, because (almost by definition) any number multiplied by 0 is 0. 0/0 is the only “division question” with multiple answers, all just as good as each other without further context.

And in contrast when you ask the question “1/0” (or 2/0, etc), there is no answer. No number yields 1 when multiplied by 0.