Mandell shows that, under some hypotheses, the $\mathbb{F}_p$-cochains detect the $\mathbb{F}_p$-homotopy type in the sense that there is such an equivalence $X \simeq Y$, if and only if, there is an equivalence of $E_\infty$-algebras $C_*(X; \mathbb{F}_p) \simeq C_*(X;\mathbb{F}_p)$. This is different than saying that $\mathbb{F}_p$-homotopy theory is equivalent to the homotopy theory of $E_\infty$-algebras in $\mathbb{F}_p$-cochain complexes because it says nothing about essential surjectivity or equivalence of mapping spaces. Nonetheless, it is still reasonable to call a space $p$-formal if there is an equivalence of $E_\infty$ algebras $C_*(X;\mathbb{F}_p) \simeq H_*(X;\mathbb{F}_p)$, and this is a useful, if rarely satisfied, condition. As an aside, Mandell showed this result actually be improved to an equivalence of homotopy theories if one instead takes coefficients in the $p$-adics$\bar{\mathbb{F}}_p$.
Mandell also shows that our only guess for the Lie model of $\mathbb{F}_p$-homotopy theory, the Koszul dual of $C_*(X;\mathbb{F}_p)$ is actually contractible. I expect this result implies there is not an easy way to detect homotopy groups from the $E_\infty$-algebra model.
All is not lost! The correct way to model $p$-torsion information with Lie algebras was demonstrated by Heuts. Using chromatic homotopy theory, one can construct the $v_n$-localization of spaces and spectra. These localizations can see $p$-torsion information when $n>0$ and when $n=0$ it coincides with rationalization. Heuts showed that $v_n$-local spaces are modeled by Lie algebras in $v_n$-local spectra. I don't think there is a definition of coformality in this context, but I would be very interested to see one.