We propose and characterize a protocol for rapid near-ground state cooling of the center-of-mass (c.m.) mode of a large trapped ion crystal. When the initial mean thermal occupation of the mode ${\bar{n}}_i$ is small compared to the number of ions $N$ (${\bar{n}}_i\ll N$), a red sideband drive implements a beam-splitter type state exchange (SWAP operation) between the mode and the collective spin of the $N$ ions, with the latter effectively serving as a quantum harmonic oscillator. Subsequently, a reset of the spins removes the entropy, leading to near-ground state cooling of the c.m. mode. We term this protocol as quantum beam-splitter cooling (QBSC). We analyze the impact of several practical imperfections on the final temperature achievable under QBSC, including finite ion number, off-resonant carrier and blue-sideband contributions, the sideband drives arising from spectator modes, and spatially inhomogeneous driving across the crystal. We outline practical strategies to eliminate the carrier drive. We extend our analysis to multiple cycles of QBSC and also generalize the analysis to demonstrate the cooling of non-c.m. modes using this protocol. Furthermore, we show that measuring the population statistics of the ions at the end of the SWAP operation can enable near-optimal quantum beam-splitter thermometry (QBST), with the classical Fisher information approaching the quantum Fisher information of a thermal state. We discuss the connection of QBSC with continuous sideband cooling and compare QBST with a recently proposed rapid adiabatic passage-based thermometry scheme. Our work constitutes an example of harnessing many-body effects to open new routes to laser cooling and thermometry in large trapped ion crystals.