“Zero people” and “needing” is an oxymoron, nothing (zero) can’t be associated to any natural thing (like “needing”), that’s what I was trying to say in my not so clear comment above.
Writing this down I realized we’re still trying to write in fancier words what the pre-Socratics had a clear understanding of 2500+ years ago, and speaking of the Greeks is too bad that Wolfram didn’t mention Plato by name in the first few paragraphs, the chair example is practically taken from him (expecting a Parmenides quote would have probably been too much).
I think I understand what you're trying to say. Let me try to motivate algebra in less explicitly algebraic terms for you:
Zero is an algebraic concept of nothing. While it refers to no physical thing, its existence in algebra is necessary to describe several mathematical laws, and several properties it has in algebra inherently derives from its algebraic concept of nothing.
Let's define both addition and multiplication [1]. Addition is an abstract representation of combination: you combine a pile of 2 things and a pile of 3 things to get a pile 5 things. Now zero is the model for what you can combine with anything else that doesn't do anything: combining a pile of 0 things (that is, nothing) with a pile of 3 things leaves you a pile of 3 things. Negative numbers represent undoing a combination: combine a pile of 5 things with a pile of -3 things (i.e., "take 3 things from the pile") leaves with a pile of 2 things. Negative numbers and zero numbers may not necessarily have a direct physical analogue, but by introducing them for algebraic purposes, things actually become simpler: you use the same terminology and logic to deal with both adding and removing things, or perhaps coming to the conclusion that in the end there's no net effect.
Now, multiplication is scaling. A half a pile of 2 things is 1 thing. Scaling and combining interact with each other, too. Taking half of a pile of 3 things and half of a pile of 5 things is the same taking half of a pile of 8 things. Or I can say that doubling a pile and then adding another of the original is the same as tripling a pile (i.e., 2x + x = 3x).
This is where things get interesting. We can do nothing by adding a pile of something and immediately taking it away, leaving us with what we started (i.e., 0 = a·x - a·x). From above, we can also see that that is the same as adding a pile whose scale is 0 (i.e., a·x - a·x = (a - a)·x). Simplifying the equation a bit, we end up with 0 = 0·x: multiplying by 0 must yield 0 to make both addition and multiplication make sense. So the concept of nothing times anything yielding nothing isn't a requirement of nothing itself, but it's a requirement of how addition and multiplication works, and how nothing itself interacts with those operations.
Incidentally, the deeper you dive into mathematics, the more important you realize the concept of 0--of nothing--actually is. The most powerful ways to describe operations are based on how they arrive at doing nothing in interestingly nontrivial ways. And things that don't have ways to do nothing tend not to be very interesting structures to look at.
[1] I'm alluding to a vector spaces here, although by glossing over the difference between scalars and vectors, it could also be viewed as rings instead.
Writing this down I realized we’re still trying to write in fancier words what the pre-Socratics had a clear understanding of 2500+ years ago, and speaking of the Greeks is too bad that Wolfram didn’t mention Plato by name in the first few paragraphs, the chair example is practically taken from him (expecting a Parmenides quote would have probably been too much).