That is false. I am taking a break from something else, so I will mostly refer to other MO answers. I gather from your example that you allow $X$ to be singular. Then Simpson proved that every finitely presented group $G$ is isomorphic to the fundamental group of a (usually singular) complex projective variety $X$, Theorem 12.1 of the following.

MR2918179

Simpson, Carlos

Local systems on proper algebraic V-manifolds.

Pure Appl. Math. Q. 7 (2011), no. 4,

Special Issue: In memory of Eckart Viehweg, 1675–1759.

I found this reference from Andy Putman's answer to this MO question.

Fundamental groups of compact Kähler manifolds

Let $G$ be a nontrivial finitely presented group whose Abelianization is trivial, e.g., the alternating group $\mathfrak{A}_n$ for $n\geq 5$. Then $\text{Pic}^0(X)$ is trivial. Thus the structure sheaf of $X$ has the invariance property. Yet the structure sheaf is not the pushforward of a coherent sheaf, because the rank of the structure sheaf, $1$, is not divisible by the order of $G$.

Let me anticipate the next question: "What if $E_\rho\otimes F$ is isomorphic to $F^{\oplus r}$ for every complex representation $\rho$ of $\pi_1(X)$ with finite dimension $r$ and with associated vector bundle $E_\rho$ on $X$?" However, the fantastic answers to the following question resolve this negatively; there are many finitely presented groups having no nontrivial representation, and then you can take $F$ to be the structure sheaf.

Finitely presented sub-groups of GL(n,C)

**Edit.** I just noticed the edit by the OP that allows (and insists) that $X$ be highly reducible. That makes things much simpler. Let $<g_1,\dots,g_m|p_1,\dots, p_n>$ be a presentation of $G$. For each generator $g_i$, let $X'_i \cong C_i\times \mathbb{P}^3$ be an irreducible component, where $(C_i,x_i)$ is a nodal plane cubic with a marked point $p_i$. Let $(C_0,y_1,\dots,y_m)$ be an $m$-marked curve of genus $0$, and let $X'_0$ be $C_0\times \mathbb{P}^3$.

Begin with $X'$, the reducible $4$-fold obtained by gluing each $X'_i,$ $i=1,\dots,m$, to $X'_0$ by identifying $\{x_i\}$ with $\{y_i\}$, inducing an identification of $\{x_i\}\times \mathbb{P}^3$ with $\{y_i\}\times \mathbb{P}^3$. The fundamental group of $X'$ is the free group on $m$-generators $<g_1,\dots,g_m>$.

Moreover, for every element $p$ of the fundamental group, the subgroup $<p>$ of the fundamental group is the image fundamental group under pushforward for an unramified, finite morphism of complex projective schemes, $$f_p:D_p\times \mathbb{P}^3\to X',$$ where $D_p$ is an $n$-gon of $\mathbb{P}^1$s that lifts to $\widetilde{X}'$ once we separate a single node. Choose a closed immersion, $$e_p:D_p\hookrightarrow \mathbb{P}^3.$$ The graph of $e_p$ in $D_p\times \mathbb{P}^3$ is a copy of $D_p$.

For each $j=1,\dots,n$, let $W_j$ denote the image under $f_{p_j}$ of the graph of $e_{p_j}$. This is an $n$-gon in $X'$ whose image fundamental group in $<g_1,\dots,g_m>$ equals $<p_j>$. If the embeddings $e_{j}$ are chosen sufficiently general, then these curves $W_j$ are pairwise disjoint. Let $X_j$ be a copy of $\mathbb{P}^3$, and glue this to $X'$ by identifying $W_j$ with time image of $e_{p_j}$ in $X_j$. Then the reduced, complex projective scheme $X$ obtained by gluing each $X_j$ to $X'$ has fundamental group $< g_1,\dots,g_m|p_1,\dots,p_n>$.