Here, shouldn't the $x_i$s map to $x_i + \mathfrak{q}^2$, rather than to $x_i +\mathfrak{q}$? If not, then everything would be mapped to degree $1$ elements.
This is another one:
Here, where is the fact that $k$ maps isomorphically onto $A/\mathfrak{m}$ used? If $f_s$ has coefficients in $\mathfrak{m}$, they can't be invertible and hence can only be zero as all the other elements of $k$ are invertible($k$ is a field).