We investigate the interplay between the Kondo effect and randomness in ${M}_{x}{\text{TiSe}}_{2}$ ($M=\text{Co}$, Ni, and Fe) single crystals. Although the typical low-$T$ upturn of resistivity implies the Kondo effect around the single-ion Kondo temperature ${T}_{K}$, positive magnetoresistance linearly proportional to the magnetic field and the power-law scaling of magnetization suggest the forbidden coexistence between Kondo effect and time-reversal symmetry breaking. This puzzling result is resolved by the Griffiths scenario---disorder-induced distribution of the Kondo temperature produces an effective Kondo temperature $({\overline{T}}_{K})$ much lower than ${T}_{K}$, allowing unscreened local moments above ${\overline{T}}_{K}$ and resulting in non-Fermi-liquid properties in ${M}_{x}{\text{TiSe}}_{2}$ below the percolation threshold $(x<{x}_{\text{c}})$.