The connection between the nature of a protoplanetary disk and that of a debris disk is not well understood. Dust evolution, planet formation, and disk dissipation likely play a role in the processes involved. We aim to reconcile both manifestations of dusty circumstellar disks through a study of optically thin Class III disks and how they correlate to younger and older disks. In this work, we collect literature and ALMA archival millimeter fluxes for 85 disks (8%) of all Class III disks across nearby star-forming regions. We derive millimeter-dust masses $M_{text{dust}}$ and compare these with Class II and debris disk samples in the context of excess infrared luminosity, accretion rate, and age. The mean $M_{text{dust}}$ of Class III disks is $0.29 pm 0.19~M_{oplus}$. We propose a new evolutionary scenario wherein radial drift is very efficient for non-structured disks during the Class II phase resulting in a rapid decrease of $M_{text{dust}}$. However, we find long infrared protoplanetary disk timescales of ${sim}$8~Myr, which are consistent with overall slow disk evolution. In structured disks, the presence of dust traps allows for the formation of planetesimal belts at large radii, such as those observed in debris disks. We propose therefore that the planetesimal belts in debris disks are the result of dust traps in structured disks, whereas protoplanetary disks without dust traps decrease in dust mass through radial drift and are therefore undetectable as debris disks after the gas has dissipated. These results provide a hypothesis for a novel view of disk evolution.
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