Evolution of the radial ISM metallicity gradient in the Milky Way disk since redshift $\approx 3$

Recent works identified a way to recover the time evolution of a galaxy's disk metallicity gradient from the shape of its age--metallicity relation. However, the success of the method is dependent on how the width of the star-forming region evolves over time, which in turn is dependent on a galaxy's present-day bar strength. In this paper, we account for the time variation in the width of the star-forming region when deriving the interstellar medium (ISM) metallicity gradient evolution over time ($\rm \nabla [Fe/H](\tau)$), which provides more realistic birth radii estimates of Milky Way (MW) disk stars. Using MW/Andromeda analogues from the TNG50 simulation, we quantified the disk growth of newly born stars as a function of present-day bar strength to provide a correction that improves recovery of $\rm \nabla [Fe/H](\tau)$. In TNG50, we find that our correction reduces the median absolute error in recovering $\rm \nabla [Fe/H] (\tau)$ by over 30%. To confirm its universality, we test our correction on two galaxies from NIHAO-UHD and find the median absolute error is over 3 times smaller even in the presence of observational uncertainties for the barred, MW-like galaxy. Applying our correction to APOGEE DR17 red giant MW disk stars suggests the effects of merger events on $\rm \nabla [Fe/H](\tau)$ are less significant than originally found, and the corresponding estimated birth radii expose epochs when different migration mechanisms dominated. Our correction to account for the growth of the star-forming region in the disk allows for better recovery of the evolution of the MW disk's ISM metallicity gradient and, thus, more meaningful stellar birth radii estimates. With our results, we are able to suggest the evolution of not only the ISM gradient, but also the total stellar disk radial metallicity gradient, providing key constraints to select MW analogues across redshift.