  The postulate of gauge invariance in nature does not lend itself directly to
implementations of lattice gauge theories in modern setups of quantum synthetic
matter. Unavoidable gauge-breaking errors in such devices require gauge
invariance to be enforced for faithful quantum simulation of gauge-theory
physics. This poses major experimental challenges, in large part due to the
complexity of the gauge-symmetry generators. Here, we show that gauge
invariance can be reliably stabilized by employing simplified \textit{local
pseudogenerators} designed such that within the physical sector they act
identically to the actual local generator. Dynamically, they give rise to
emergent exact gauge theories up to timescales polynomial and even exponential
in the protection strength. This obviates the need for implementing often
complex multi-body full gauge symmetries, thereby further reducing experimental
overhead in physical realizations. We showcase our method in the $\mathbb{Z}_2$
lattice gauge theory, and discuss experimental considerations for its
realization in modern ultracold-atom setups.
