I think the question is not so much what we can do without zero, but how zero could remain undiscovered when humans begin to advance.
One of my favourite quotes from one of my favourite contemporary mathematicians (Roger Penrose) is that it's always possible to create an equations from numbers of a given type whose answer falls beyond that type.:
Positive Integers, 1-1=a or 1-6=a
All Integers, 1/2=a
Rationals, circumference of circle/diameter of circle =a
Irrationals, sqrt(-1)=a
- Positive Integers: $1-1=a$ or $1-6=a$
- All Integers: $\frac12=a $
- Rationals: $\sqrt2 =a$
- Irrationals: $\sqrt{-1}=a$
Before science, the primary driver of mathematics was commerce. How can we possibly pay off an account if there is no concept of zero? How do we record balances as fully paid?
It's not so much that we couldn't advance without the concept of zero, it's more that discovering the concept of zero was always going to be a byproduct of advancement. Conceptually, it was always going to appear in mathematics because it's a necessary concept on which we build additional foundations.
I'd argue that the Greeks' initial struggle with zero as a concept only pushed back western civilisation several centuries, rather than truly impeding it. When it was (re)introduced to European culture during the middle ages by the Spanish Moors, it was embraced as a necessity. That Egypt (for example) had the concept of zero nearly 2 millenia before the time of Jesus should identify it as a concept that was always inevitable.
