The origin of high-energy emission in blazars jets (i.e., leptonic versus hadronic) has been a longstanding matter of debate. Here, we focus on one variant of hadronic models where proton synchrotron radiation accounts for the observed steady {gamma}-ray blazar emission. Using analytical methods, we derive the minimum jet power (P_j,min_) for the largest blazar sample analyzed to date (145 sources), taking into account uncertainties of observables and jet's physical parameters. We compare P_j,min_ against three characteristic energy estimators for accreting systems, i.e., the Eddington luminosity, the accretion disk luminosity, and the power of the Blandford-Znajek process, and find that P_j,min_ is about 2 orders of magnitude higher than all energetic estimators for the majority of our sample. The derived magnetic field strengths in the emission region require either large amplification of the jet's magnetic field (factor of 30) or place the {gamma}-ray production site at sub-pc scales. The expected neutrino emission peaks at ~0.1-10EeV, with typical peak neutrino fluxes ~10^-4^ times lower than the peak {gamma}-ray fluxes. We conclude that if relativistic hadrons are present in blazar jets, they can only produce a radiatively subdominant component of the overall spectral energy distribution of the blazar's steady emission.