Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $ \mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which yields an fully faithful embedding of tensor $ \mathbb{Q}$-categories $CHM_k\rightarrow DM_k$, where $CHM_k$ is athe category of Chow motives with $\mathbb{Q}$-coefficients over $k$.

For $\ell$ prime to the characteristic of $k$ one has the $\ell$-adic realisation functor $r_{\ell}:DM_k\rightarrow D^b(Vec_{\mathbb{Q}\ell})$. As I understand it, if $X$ is smooth projective variety over $k$ then the cohomology of $r_{\ell}(M(X))$ is just the $\ell$-adic cohomology of $X$.

Now the conjectural motivic $t$-structure on $DM_k$ has the property that the realisation functor $r_{\ell}$ is $t$-exact.Thus those objects in the heart of this $t$-structure (the conjectural category of mixed motives $MM_k$) have realisation with trivial cohomology outside of degree zero.

Here is what is confusing me: for a general smooth projective variety its $\ell$-adic cohomology is not always concentrated in degree zero. Thus for such $X$, the motive $M(X)$ is not in the category of mixed motives. This can't be correct as the category of mixed motives should contain the category of pure motives. Where am I going wrong?