Doppler-broadening thermometry (DBT) can be used as a calibration-free primary thermometer suitable for practical applications, e.g. reliably measuring temperatures over long periods of time in environments where sensor retrieval for recalibration is impractical. We report on our proof-of-concept investigations into DBT with alkali-metal-vapour cells, with a particular focus on both absorption and frequency accuracy during scans. We reach sub-kelvin temperature accuracy, and experimental absorption-fit residuals below 0.05%, in a simple set-up. The outlook for portable, practical devices is bright, with clear prospects for future improvement. This article is part of the Theo Murphy meeting issue textquoteleftThe redefined kelvin: progress and prospectstextquoteright.