The energy released by active galactic nuclei (AGN) has the potential to heat or remove the gas of the ISM, thus likely impacting the cold molecular gas reservoir of host galaxies at first, with star formation following as a consequence on longer timescales. Previous works on high-z galaxies, which compared the gas content of those without identified AGN, have yielded conflicting results, possibly due to selection biases and other systematics.To provide a reliable benchmark for galaxy evolution models at cosmic noon (z=1-3), two surveys were conceived: SUPER and KASHz, both targeting unbiased X-ray-selected AGN at z>1 that span a wide bolometric luminosity range. In this paper we assess the effects of AGN feedback on the molecular gas content of host galaxies in a statistically robust, uniformly selected, coherently analyzed sample of AGN at z=1-2.6, drawn from the KASHz and SUPER surveys. By using targeted and archival ALMA data in combination with dedicated SED modeling, we retrieve CO and far-infrared (FIR) luminosity as well as M* of SUPER and KASHz host galaxies. We selected non-active galaxies from PHIBBS, ASPECS, and multiple ALMA/NOEMA surveys of submillimeter galaxies in the COSMOS, UDS, and ECDF fields. By matching the samples in redshift, stellar mass, and FIR luminosity, we compared the properties of AGN and non-active galaxies within a Bayesian framework.We find that AGN hosts at given FIR luminosity are on average CO depleted compared to non-active galaxies, thus confirming what was previously found in the SUPER survey. Moreover, the molecular gas fraction distributions of AGN and non-active galaxies are statistically different, with the distribution of AGN being skewed to lower values. Our results indicate that AGN can indeed reduce the total cold molecular gas reservoir of their host galaxies. Lastly, by comparing our results with predictions from three cosmological simulations (TNG, Eagle, and Simba) filtered to match the properties of observed AGN, AGN hosts, and non- active galaxies, we confirm already known discrepancies and highlight new discrepancies between observations and simulations.