We estimate the age of the Universe using the Xiang & Rix sample of 247,103 Milky Way stars with high-resolution spectroscopy from LAMOST DR7 and $Gaia$ eDR3 parallaxes. Stellar ages were estimated using YY isochrones up to 20 Gyr. To remove stars with unusually high and precise ages, we require old stars to be metal-poor and $α$-enriched. We also require consistency between YY ages and those obtained with FLAME based only on $Gaia$ data. Our final sample of 155,600 stars within 5 kpc provides consistent cosmic age estimates using several techniques of increasing rigour. Our main results use an MCMC reconstruction of the latent age distribution, though our iterative reconstruction is very similar. Applying an innovative approach to our MCMC reconstruction and its uncertainties, we find that the oldest star has an age of $A_\star = 13.73^{+0.18}_{-0.15}$ Gyr. Varying the quality cuts can at most reduce this to $A_\star = 13.31^{+0.21}_{-0.18}$ Gyr or raise it to $14.02^{+0.18}_{-0.15}$ Gyr using a much lower or higher age-dependent metallicity ceiling, respectively. Our inferred $A_\star$ is consistent with the 13.6 Gyr expected in CMB-calibrated $Λ$CDM, assuming the first long-lived stars formed when the Universe was 0.2 Gyr old. This agreement casts doubt on solutions to the Hubble tension solely through new physics prior to recombination, which generally imply a cosmic age of $12.9 \pm 0.2$ Gyr to match low redshift probes. It is difficult for stellar modelling uncertainties to reconcile such a low age with our result given the low metallicities of the oldest stars in our sample and independent asteroseismic constraints.