The Auger Engineering Radio Array (AERA) is an array of $153$ radio antennas spanning an area of $17$ km$^2$, currently the largest of its kind, that probes the nature of ultra-high energy cosmic rays at energies around the transition from Galactic to extra-galactic origin. It measures the MHz radio emission of extensive air showers produced by cosmic rays hitting our atmosphere. The elemental composition of cosmic rays is a crucial piece of information in determining what the sources of cosmic rays are and how cosmic rays are accelerated. This composition can be obtained from the mass-sensitive parameter $X_\mathrm{max}$, the depth of the shower maximum. We reconstruct $X_\mathrm{max}$ with a likelihood analysis by comparing the measured radio footprint on the ground to an ensemble of footprints from Monte-Carlo CORSIKA/CoREAS air shower simulations. We compare our $X_\mathrm{max}$ reconstruction with fluorescence $X_\mathrm{max}$ measurements on a per-event basis, a setup unique to the Pierre Auger Observatory, and show the methods to be compatible. Furthermore, we extensively validate our reconstruction by identifying and correcting for systematic uncertainties. We determine the resolution of our method as a function of energy and reach a precision better than $15$ gcm$^{-2}$ at the highest energies. With a bias-free set of around $600$ showers, we find a light to light-mixed composition at energies between $10^{17.5}$ to $10^{18.8}$ eV, also in agreement with Auger fluorescence measurements.