We propose locally stable sparse hard-disk packings, as introduced by B\"or\"oczky, as a model for the analysis and benchmarking of Markov-chain Monte Carlo (MCMC) algorithms. We first generate such packings in a square box with periodic boundary conditions and analyze their properties. We then study how local MCMC algorithms, namely the Metropolis algorithm and several versions of event-chain Monte Carlo (ECMC), escape from configurations that are obtained by slightly reducing all disk radii by a relaxation parameter. A scaling analysis is confirmed by simulation results. We obtain two classes of ECMC, one in which the escape time varies algebraically with the relaxation parameter (as for the local Metropolis algorithm) and another in which the escape time scales as the logarithm of the relaxation parameter. We discuss the connectivity of the hard-disk sample space, the ergodicity of local MCMC algorithms, as well as the meaning of packings in the context of the NPT ensemble. Our work is accompanied by open-source, arbitrary-precision software for B\"or\"oczky packings (in Python) and for straight, reflective, forward, and Newtonian ECMC (in Go).