Associative Multivariate Permutation
A couple of papers by Faugère and Perret (and, in one case, von zur Gathen) provide efficient (i.e., polynomial time) algorithms to solve the Functional Decomposition Problem:


Given: a list of $u$ multivariate polynomials $H = (h_1, \ldots, h_u) \in (\mathbb{K}[x_1, \ldots, x_n])^u$ in $n$ variables over some field $\mathbb{K}$.
Task: find a non-trivial decomposition $(F,G)$ where $F = (f_1, \ldots, f_u) \in (\mathbb{K}[x_1, \ldots, x_n])^u$ and $G = (g_1, \ldots, g_n) \in (\mathbb{K}[x_1, \ldots, x_n])^n$ such that $H = F \circ G$.


There are various definitions of "non-trivial" designed to explicitly exclude pathological cases such as where $F$ or $G$ are the identity or linear transforms. Suffice to say that any cryptosystem claiming security based on the hardness of multivariate polynomial decomposition, such as your proposals, should convincingly argue why the algorithms of Faugère et al. do not apply.

Specifically, these algorithms enable the attacker to obtain $f$ from $p$ in your proposals, even without using $g$. This breaks the schemes without inverting anything because $f$ represents the secret key.

There was a similar proposal called 2R$^-$. It was the original motivation for Faugère and Perret to study the FDP, which lead to that scheme's demise.