  We introduce a method for quantum simulation of U$(1)$ lattice gauge theories
coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent
optical lattices. The proposal enables the study of both gauge and
fermionic-matter fields without integrating out one of them in one and two
dimensions. We focus on a realistic and robust implementation that utilizes the
long-lived metastable clock state available in alkaline-earth(-like) atomic
species. Starting from an $ab\,initio$ modelling of the experimental setting,
we systematically carry out a derivation of the target U$(1)$ gauge theory.
This approach allows us to identify and address conceptual and practical
challenges for the implementation of lattice gauge theories that - while
pivotal for a successful implementation - have never been rigorously addressed
in the literature: those include the specific engineering of lattice potentials
to achieve the desired structure of Wannier functions, and the subtleties
involved in realizing the proper separation of energy scales to enable
gauge-invariant dynamics. We discuss realistic experiments that can be carried
out within such a platform using the fermionic isotope $^{173}$Yb, addressing
via simulations all key sources of imperfections, and provide concrete
parameter estimates for relevant energy scales in both one- and two-dimensional
settings.
