One immediate sufficient condition for $\{P_1,\dots, P_{M_n}\}\subset \operatorname{Hull}\{P_{Y^n|X^n=x^n}| x^n\in \mathcal{X}^n\}$
is that if for some $c>0$ then for each $m$ and $\ell\neq m$ taking $Y_m\sim P_m$ gives
$$\mathbb{P}(P_m(Y_m) \leq c\cdot P_{\ell}(Y_m))=\mathcal{o}(1/M_n).$$

Then this will be a working decoder: "*Decide $m$ was sent if $\mathbb{P}(P_m(Y_m)>c\cdot P_{\ell}(Y_m))$ for each $\ell\neq m$.*" For $c=1$ this is a maximum likelihood decoder.

This is a kind of unsatisfying answer since I am suspicious that some channel has a codebook that can be decoded, but any working decoder has to operate in a nontrivially different way than this.