Question about global quotient stacks

In "Brauer groups and quotient stacks", Edidin et. al prove the following theorem:

Theorem 2.7. Let $mathcal{X}$ be an algebraic stack over a Noetherian base (of finite type). Then the diagonal $mathcal{X}to mathcal{X}times mathcal{X}$ is quasi-finite if and only if there is a finite surjective morphism $Xto mathcal{X}$ for a scheme $X$.

On the other hand, Kresch in "Cycle groups for Artin stacks" proves the following:

Proposition 3.5.7. Let $mathcal{X}$ be a stack of finite type over a field. The the following are equivalent:
1) The diagonal is quasi-finite;
2) The stabilizer $mathcal{X}times_{mathcal{X}timesmathcal{X}}mathcal{X}to mathcal{X}$ is quasi-finite.
Further, if $mathcal{X}$ has quasi-finite diagonal $mathcal{X}$ admits a stratification by quotient stacks.

Now, suppose that $mathcal{X}$ is already a quotient stack $[Y/G]$, say with $Y$ an affine scheme and $G$ some group scheme (both of finite type over a field). Then $mathrm{id}: Ytomathcal{X}$ is a finite surjective morphism, so by 2.7 above have quasi-finite diagonal. Then by 3.5.7 the stabilizer is quasi-finite, but this seems false in general. For instance, take $G=GL(n)$ and then you are almost guaranteed to have non-finite stabilizers.

What am I missing here? It's obvious that there's something here that I've gotten wrong.