We present an extended zero-field muon spin relaxation (ZF-$mu$SR) study of overdoped Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+delta}$ (Bi2212) single crystals, intended to elucidate the origin of weak quasistatic magnetism previously detected by $mu$SR in the superconducting and normal states of optimally-doped and overdoped samples. New results on heavily-overdoped single crystals show a similar monotonically decreasing ZF-$mu$SR relaxation rate with increasing temperature that persists above the pseudogap (PG) temperature $T^*$ and does not evolve with hole doping ($p$). Additional measurements using an ultra-low background apparatus confirm that this behavior is an intrinsic property of Bi2212, which cannot be due to magnetic order associated with the PG phase. Instead we show that the temperature-dependent relaxation rate is most likely caused by structural changes that modify the contribution of the nuclear dipole fields to the ZF-$mu$SR signal. Our results for Bi2212 emphasize the importance of not assuming the nuclear-dipole field contribution is independent of temperature in ZF-$mu$SR studies of high-temperature (high-$T_c$) cuprate superconductors, and do not support a recent $mu$SR study of YBa$_2$Cu$_3$O$_{6+x}$ that claims to detect magnetic order in the PG phase.
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