The easiest counterexample is $p=2, q=0$. In this case the usual motivic cohomology vanishes, and if $X$ is normal $$H^2_{et}(X,Z)=H^1_{et}(X,Q/Z)= Hom(\pi_1(X),Q/Z)$$ which does not vanish even for fields (which are not separably closed).

Another example is and p=3, q=1. In this case you get the Brauer group which doesn't vanish in general.

The easiest counterexample is $p=2, q=0$. In this case the usual motivic cohomology vanishes, and if $X$ is normal $$H^2_{et}(X,\mathbf{Z})=H^1_{et}(X,\mathbf{Q}/\mathbf{Z})= \mathrm{Hom}(\pi_1^{et}(X),\mathbf{Q}/\mathbf{Z})$$ which does not vanish even for fields (which are not separably closed).

Another example is $p=3, q=1$. In this case you get the Brauer group which doesn't vanish in general.

The easiest counterexample is X the spectrum of a field and p=3, q=1. In this case you get the Brauer group which doesn't vanish in general.

The easiest counterexample is $p=2, q=0$. In this case the usual motivic cohomology vanishes, and if $X$ is normal $$H^2_{et}(X,Z)=H^1_{et}(X,Q/Z)= Hom(\pi_1(X),Q/Z)$$ which does not vanish even for fields (which are not separably closed).

Another example is and p=3, q=1. In this case you get the Brauer group which doesn't vanish in general.