Kondo-based semimetals and semiconductors are of extensive current interest as a viable platform for strongly correlated states in the dilute carrier limit. It is thus important to explore the routes to understand such systems. One established pathway is through the Kondo effect in metallic nonmagnetic analogs, in the so called half-filling case of one conduction electron and one 4 f electron per site. Here, we demonstrate that Kondo-based semimetals develop out of conduction electrons with a low-carrier density in the presence of an even number of rare-earth sites. We do so by studying the Kondo material Yb₃Ir₄Ge₁₃ along with its closed-4f-shell counterpart, Lu₃Ir₄Ge₁₃ . Through magnetotransport, optical conductivity, and thermodynamic measurements, we establish that the correlated semimetallic state of Yb₃Ir₄Ge₁₃ below its Kondo temperature originates from the Kondo effect of a low-carrier conduction-electron background. In addition, it displays fragile magnetism at very low temperatures, which in turn, can be tuned to a Griffiths-phase-like regime through Lu-for-Yb substitution. These findings are connected with recent theoretical studies in simplified models. Our results can pave the way to exploring strong correlation physics in a semimetallic environment.