We present the first results of the Galaxy Activity, Torus, and Outflow Survey (GATOS), a project aimed at understanding the properties of the dusty molecular tori and their connection to the host galaxy in nearby Seyfert galaxies. Our project expands the range of active galactic nuclei (AGN) luminosities and Eddington ratios covered by previous surveys of Seyferts conducted by the Atacama Large Millimeter Array (ALMA), allowing us to study the gas feeding and feedback cycle in a combined sample of 19 Seyferts. We used ALMA to obtain new images of the emission of molecular gas and dust using the CO(3-2) and HCO+(4-3) lines as well as their underlying continuum emission at 870 μm with high spatial resolutions (0.1″ ∼ 7 − 13 pc) in the circumnuclear disks (CND) of ten nearby (D < 28 Mpc) Seyfert galaxies selected from an ultra-hard X-ray survey. Our new ALMA observations detect 870 μm continuum and CO line emission from spatially resolved disks located around the AGN in all the sources. The bulk of the 870 μm continuum flux can be accounted for by thermal emission from dust in the majority of the targets. For most of the sources, the disks show a preponderant orientation perpendicular to the AGN wind axes, as expected for dusty molecular tori. The median diameters and molecular gas masses of the tori are ∼42 pc and ∼6 × 105 M⊙, respectively. We also detected the emission of the 4-3 line of HCO+ in four GATOS targets. The order of magnitude differences found in the CO/HCO+ ratios within our combined sample point to a very different density radial stratification inside the dusty molecular tori of these Seyferts. We find a positive correlation between the line-of-sight gas column densities responsible for the absorption of X-rays and the molecular gas column densities derived from CO toward the AGN in our sources. Furthermore, the median values of both column densities are similar. This suggests that the neutral gas line-of-sight column densities of the dusty molecular tori imaged by ALMA significantly contribute to the obscuration of X-rays. The radial distributions of molecular gas in the CND of our combined sample show signs of nuclear-scale molecular gas deficits. We also detect molecular outflows in the sources that show the most extreme nuclear-scale gas deficits in our sample. These observations find for the first time supporting evidence that the imprint of AGN feedback is more extreme in higher luminosity and/or higher Eddington ratio Seyfert galaxies.