Using the Westerbork Synthesis Radio Telescope, we carried out shallow H i absorption observations of a flux-selected (S1.4 GHz > 50 mJy) sample of 93 radio active galactic nuclei (AGN), which have available SDSS (Sloan Digital Sky Survey) redshifts between 0.02 <z <0.23. Our main goal is to study the gas properties of radio sources down to S1.4 GHz flux densities not systematically explored before using, for the first time, stacking of absorption spectra of extragalactic H i. Despite the shallow observations, we obtained a direct detection rate of ~29%, comparable with deeper studies of radio galaxies. Furthermore, detections are found at every S1.4 GHz flux level, showing that H i absorption detections are not biased toward brighter sources. The stacked profiles of detections and non-detections reveal a clear dichotomy in the presence of H i, with the 27 detections showing an average peak τ = 0.02 corresponding to N(H i) ~(7.4 ± 0.2) × 1018 (Tspin/cf) cm-2, while the 66 non-detections remain undetected upon stacking with a peak optical depth upper limit τ <0.002 corresponding to N(H i) <(2.26 ± 0.06) × 1017 (Tspin/cf) cm-2 (using a FWHM of 62 kms-1, derived from the mean width of the detections). Separating the sample into compact and extended radio sources increases the detection rate, optical depth, and FWHM for the compact sample. The dichotomy for the stacked profiles of detections and non-detections still holds between these two groups of objects. We argue that orientation effects connected to a disk-like distribution of the H i can be partly responsible for the dichotomy that we see in our sample. However, orientation effects alone cannot explain all the observational results, and some of our galaxies must be genuinely depleted of cold gas. A fraction of the compact sources in the sample are confirmed by previous studies as likely young radio sources (compact steep spectrum and gigahertz peaked spectrum sources). These show an even higher detection rate of 55%. Along with their high integrated optical depth and wider profile, this reinforces the idea that young radio AGN are embedded in a medium that is rich in atomic gas. Part of our motivation is to probe for the presence of faint H i outflows at low optical depth using stacking. However, the stacked profiles do not reveal any significant blueshifted wing. We are currently collecting more data to investigate the presence of outflows. The results presented in this paper are particularly relevant for future surveys in two ways. The lack of bias toward bright sources is encouraging for the search for H i in sources with even lower radio fluxes planned by such surveys. The results also represent a reference point when searching for H i absorption at higher redshifts.