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We study a Su-Schrieffer-Heeger electron-phonon model on a square lattice by means of auxiliary-field quantum Monte Carlo simulations. The addition of a symmetry-allowed interaction permits analytical integration over the phonons at the expense of discrete Hubbard-Stratonovich fields with imaginary-time correlations. Using single-spin-flip updates, we investigate the phase diagram at the O(4)-symmetric point as a function of hopping $t$ and phonon frequency $\omega_0$. For $t=0$, where electron hopping is boson-assisted, the model maps onto an unconstrained $\mathbb{Z}_2$ gauge theory. A key quantity is the emergent effective flux per plaquette, which equals $\pi$ in the assisted-hopping regime and vanishes for large $t$. Phases in the former regime can be understood in terms of instabilities of emergent Dirac fermions. Our results support a direct and continuous transition between a $(\pi,0)$ valence bond solid (VBS) and an antiferromagnetic (AFM) phase with increasing $\omega_0$. For large $t$ and small $\omega_0$, we find finite-temperature signatures of a previously reported $(\pi,\pi)$ VBS ground state related to a nesting instability. With increasing $\omega_0$, AFM order again emerges., Comment: 18 pages, 18 figures