In this paper, Gilton and Krueger show that the failure of approachability at $\omega_2$ is implied by the existence of a disjoint stationary sequence, which is a disjoint sequence of sets $\langle S_\alpha : \alpha \in T \rangle$, where $T \subseteq \omega_2 \cap \mathrm{cof}(\omega_1)$ is stationary and each $S_\alpha$ is a stationary subset of $[\alpha]^\omega$. It follows from the arguments in their paper that Mitchell's forcing adds a disjoint stationary sequence. They key is that the quotient forcings over initial segments are proper (projections of ccc x countably closed, just like the whole forcing).
Now in the extension by $\mathbb C \times \mathbb T$ like in the question, the approachability property holds because CH holds in an inner model with the same cardinals (the extension by $\mathbb T$). Thus the passage from $V^{\mathbb M}$ to $V^{\mathbb C \times \mathbb T}$ must kill the disjoint stationary sequence $\langle S_\alpha : \alpha \in T \rangle$. Since it is $\kappa = \omega_2$-c.c., it cannot kill the stationarity of $T$. Thus it must kill almost all of the $S_\alpha$. Since they are stationary subsets of countable subsets of ordinals of size $\omega_1$, they correspond to stationary subsets of $\omega_1$, so the quotient kills many stationary subsets of $\omega_1$.
EDIT: A more direct argument, without using all of the Gilton-Krueger work. The stationary subset of $[\alpha]^\omega$ referred to above is just $$([\alpha]^\omega)^{V[G \restriction \alpha][c_\alpha]} \setminus ([\alpha]^\omega)^{V[G \restriction \alpha]},$$ where $c_\alpha$ is the Cohen real added at stage $\alpha$. By a theorem of Gitik, this is stationary, and its stationarity is preserved by the tail forcing. But in the model $V^{\mathbb T}$, there is a club in $[\alpha]^\omega$ consisting of countable sets from $V$.