  Few layers of graphene at small twist-angles have emerged as a fascinating
platform for studying the problem of strong interactions in regimes with a
nearly quenched single-particle kinetic energy and non-trivial band topology.
Starting from the strong-coupling limit of twisted bilayer graphene with a
vanishing single-electron bandwidth and interlayer-tunneling between the same
sublattice sites, we present an ${\it exact}$ analytical theory of the Coulomb
interaction-induced low-energy optical spectral weight at all ${\it integer}$
fillings. In this limit, while the interaction-induced single-particle
dispersion is finite, the optical spectral weight vanishes identically at
integer fillings. We study corrections to the optical spectral weight by
systematically including the effects of experimentally relevant strain-induced
renormalization of the single-electron bandwidth and interlayer tunnelings
between the same sublattice sites. Given the relationship between the optical
spectral weight and the diamagnetic response that controls superconducting
$T_c$, our results highlight the relative importance of specific parent
insulating phases in enhancing the tendency towards superconductivity when
doped away from integer fillings.
