How is the defintion of the mean preserving spread (MPS) not too general?
I realized my mistake whilst writing this, so I will just share the answer as well.

Essentially my mistake was, that I assumed this aspect of the definition ($E[z|x_g] = 0\ \forall\ x_g$) to mean that the expected value of z has to be 0 given, that we are in the lottery that is given by the random variable $x_g$. This reading was however wrong. It actually means, that the expected value of z has to be 0, given that we are at a particular value of $x_g$. This has to be individually true for all values of $x_g$.

Thus, using this definition correctly a negative spreading is not possible anymore, the MPS of the original lottery is therefore always at least as risky as the original lottery. A risk averse decision maker will thus never want to choose an MPS.