Why do we state that "$A$ vector space has a zero vector", then we prove that for any vector space, the zero vector is unique?
Why don't we say instead (as a property) that a vector space has a unique zero vector?
Why is the property being general better?
It is good mathematical strategy to posit the least-restrictive (most general) axioms. Saying there exists a zero vector is less-restrictive than specifying a specific such vector.
Example: Suppose you're defining the natural numbers. You could posit $0$ is in the set and the operation of adding $1$ to any member to get another member. That's all you need.
... OR you could (in principle) state that $0$ and $1$ are in the set and that you can add $2$ to any member to get another member. Sure... that would work too, but is needlessly complex and doesn't reveal the underlying structure as well.