Gain at given frequency from $z$-plane zero-pole plot. Two methods gives different results
I think you lost the square root somewhere in your graphical method: Considering the nominator (product of lengths of vectors from every zero given point), I end up with

$$N=2*2$$

i.e. you have two vectors of length 2 (two zeros at z=-1).

Then, for the denominator you have 

$$
D = \sqrt{(A\sin\theta)^2+(A\cos\theta-1)^2}^2=A^2-2A\cos\theta+1
$$

And finally $G=N/D$ yields the same result as your analytical solution.