Lower Bound on the Probability an LWE Matrix is Primitive
I think you're overthinking it.

Hint

The event in which $A\cdot \mathbb Z_q^m \neq \mathbb Z_q^n$ is the event in which the matrix $A$ has a rank defect, which is "contained" in the event in which (say) the first row is a linear combination of the other rows.
Now, what's the probability of the latter event?