The stellar halo as well as the system of satellites of nearby Milky Way (MW)-mass galaxies both encode information about the accretion history of the galaxy. Recent progress in constraining the massive accretions (>1:10) experienced by the MW and the Andromeda galaxy (M31) offers an opportunity to understand their contributions to the dwarf galaxy population of the Local Group, as well as to help decipher the information encoded in the satellites of nearby MW-mass galaxies in the upcoming era of the James Webb and Nancy Grace Roman Space Telescopes, and the Vera Rubin Observatory. Using zoom-in simulations of MW-mass haloes, we demonstrate that the infall of a massive progenitor is accompanied by (a) the accretion of a large number of ‘associated’ subhaloes, (b) the destruction of a number of subhaloes, and (c) the ejection of some subhaloes to large galactocentric radii. Massive accreted progenitors do not increase the total number of infalling subhaloes onto a MW-mass host, but instead focus surrounding subhaloes onto the host causing a clustering in the infall time of subhaloes. This leads to a temporary elevation in the number of subhaloes found within the virial radius of the host as well as changes in the concentration of a cumulative radial profile of subhaloes. We find that the star formation quenching times of Local Group dwarf spheroidal galaxies (105 M⊙ < M* < 107 M⊙) are clustered around the times of the most massive accretions suffered by the MW and M31. Our results imply that a) the quenching time of dwarf spheroidals is a good proxy for their infall time and that b) the clustering of the quenching times of satellites can be used to constrain the time of accretion of a nearby MW-mass galaxy’s most significant accretion event.