  Freeze-in dark matter (DM) mediated by a light ($\ll$ keV) weakly-coupled
dark-photon is an important benchmark for the emerging low-mass direct
detection program. Since this is one of the only predictive, detectable
freeze-in models, we investigate how robustly such testability extends to other
scenarios. For concreteness, we perform a detailed study of models in which DM
couples to a light scalar mediator and acquires a freeze-in abundance through
Higgs-mediator mixing. Unlike dark-photons, whose thermal properties weaken
stellar cooling bounds, the scalar coupling to Standard Model (SM) particles is
subject to strong astrophysical constraints, which severely limit the fraction
of DM that can be produced via freeze-in. While it seems naively possible to
compensate for this reduction by increasing the mediator-DM coupling,
sufficiently large values eventually thermalize the dark sector with itself and
yield efficient DM annihilation to mediators, which depletes the freeze-in
population; only a small window of DM candidate masses near the $\sim$ GeV
scale can accommodate the total observed abundance. Since many qualitatively
similar issues arise for other light mediators, we find it generically
difficult to realize a viable freeze-in scenario in which production arises
only from renormalizable interactions with SM particles. We also comment on
several model variations that may evade these conclusions.
