In short: Why does 0 X 0 = 0 and not instead be undefined?

Now to build my case.

I have grown up accepting that any number divided by zero is undefined, and I can understand the reasoning behind it. However, there is an equation that I have never understood or agreed on the answer: 0 X 0 = 0.

It's taken for granted that anything multiplied by zero is zero. However, why is it that 0 X 0 yields 0? Why should the answer not be undefined?

Here is my rationale:

I would say that "multiplied by" could be synonymous with "lots of" or "instances of", so I write the following two sentences:

If there are no instances of nothing, that means there is nothing.

If there are no instances of nothing, that means there is something but how much is not defined.

Logically, which sentence makes more sense? Does not the second?

We are always told that negating a negative, nullifies that negative, we are told that "I ain't done nothin" is bad grammar because it would really mean that the person has done something, quite opposite to the intention of the speaker.

Now I'm not trying to conflate the English language with mathematical logic, but we ought to be able to be consistent in our use of language. Otherwise, what does multiplied by really mean?

Which leads me to a follow up question - Is it universally accepted that 0 X 0 = 0 more as an accepted convention or is it a mathematical paradox?

Finally, are there any famous mathematicians who have brought up this issue and have actually sided with the case that I provide that the answer should rather be undefined than 0?

Thank you for your time.