The jets of radio AGN provide one of the most important forms of active galactic nuclei (AGN) feedback, yet considerable uncertainties remain about how they are triggered. Since the molecular gas reservoirs of the host galaxies can supply key information about the dominant triggering mechanism(s), here we present Atacama Large Millimeter/sub-millimeter Array CO(1-0) observations of a complete sample of 29 powerful radio AGN ($P_{1.4\,{\rm GHz}} \gt 10^{25}$ W Hz$^{-1}$ and $0.05 \lt z \lt 0.3$) with an angular resolution of about 2-3 arcsec (corresponding to 2-8 kpc). We detect molecular gas with masses in the range $10^{8.9} \lt M_{{\rm H}_2} \lt 10^{10.2}$ M$_\odot$ in the early-type host galaxies of ten targets, while for the other 19 sources, we derive upper limits. The detection rate of objects with such large molecular masses - $34\pm 9$ per cent - is higher than in the general population of non-active early-type galaxies (ETGs: $\lt $10 per cent). The kinematics of the molecular gas are dominated in most cases by rotating disc-like structures, with diameters up to 25 kpc. Compared with the results for samples of quiescent ETG in the literature, we find a larger fraction of more massive, more extended and less settled molecular gas structures. In most of the CO-detected sources, the results are consistent with triggering of the AGN as the gas settles following a merger or close encounter with a gas-rich companion. However, in a minority of objects at the centres of rich clusters of galaxies, the accretion of gas cooling from the hot X-ray haloes is a plausible alternative to galaxy interactions as a triggering mechanism.