  For manifolds with a distinguished asymptotically flat end, we prove a
density theorem which produces harmonic asymptotics on the distinguished end,
while allowing for points of incompleteness (or negative scalar curvature) away
from this end. We use this to improve the "quantitative" version of the
positive mass theorem (in dimensions $3\leq n\leq 7$), obtained by the last two
named authors with S.-T. Yau [LUY21], where stronger decay was assumed on the
distinguished end. We also give an alternative proof of this theorem based on a
relationship between MOTS and $\mu$-bubbles and our recent work on the
spacetime positive mass theorem with boundary [LLU21].
