  The Auger Engineering Radio Array (AERA), part of the Pierre Auger
Observatory, is currently the largest array of radio antenna stations deployed
for the detection of cosmic rays, spanning an area of $17$ km$^2$ with 153
radio stations. It detects the radio emission of extensive air showers produced
by cosmic rays in the $30-80$ MHz band. Here, we report the AERA measurements
of the depth of the shower maximum ($X_\text{max}$), a probe for mass
composition, at cosmic-ray energies between $10^{17.5}$ to $10^{18.8}$ eV,
which show agreement with earlier measurements with the fluorescence technique
at the Pierre Auger Observatory. We show advancements in the method for radio
$X_\text{max}$ reconstruction by comparison to dedicated sets of CORSIKA/CoREAS
air-shower simulations, including steps of reconstruction-bias identification
and correction, which is of particular importance for irregular or sparse radio
arrays. Using the largest set of radio air-shower measurements to date, we show
the radio $X_\text{max}$ resolution as a function of energy, reaching a
resolution better than $15$ g cm$^{-2}$ at the highest energies, demonstrating
that radio $X_\text{max}$ measurements are competitive with the established
high-precision fluorescence technique. In addition, we developed a procedure
for performing an extensive data-driven study of systematic uncertainties,
including the effects of acceptance bias, reconstruction bias, and the
investigation of possible residual biases. These results have been
cross-checked with air showers measured independently with both the radio and
fluorescence techniques, a setup unique to the Pierre Auger Observatory.
