Nuclear structure effects are essential for describing hyperfine splittings from high-precision atomic spectroscopy measurements. These effects are often parametrized by the effective or elastic Zemach radii, with their difference poorly understood. We show that the longstanding discrepancy between the effective and elastic Zemach radii in 6Li and 7Li could be resolved by taking into account nuclear polarizability effects in ab initio nuclear structure calculations. Our results demonstrate that nuclear polarizability effects, negligible in 7Li, dominate in 6Li and explain the observed significant deviation between the effective and elastic Zemach radii. Furthermore, we show that the ratios between the nuclear polarizability contributions in different nuclei are universal in the limit of closure and SU(4) symmetry of nuclear forces. In particular, the nuclear polarizability contribution in an odd-odd nucleus is enhanced by a factor of ${\mu }_p/({\mu }_p+{\mu }_n)\simeq 3$, with ${\mu }_{n,p}$ denoting the nucleon magnetic moments, compared to its odd-A isotopes. The same mechanism also explains the Zemach radius deviations observed in 2H and 3He. These findings establish nuclear polarizability as the dominant source of isotope-dependent nuclear corrections to hyperfine splitting in light atoms.