Blazars have been suggested as possible neutrino sources long before the recent IceCube discovery of high-energy neutrinos. We re-examine this possibility within a new framework built upon the blazar simplified view and a self-consistent modelling of neutrino emission from individual sources. The former is a recently proposed paradigm that explains the diverse statistical properties of blazars adopting minimal assumptions on blazars' physical and geometrical properties. This view, tested through detailed Monte Carlo simulations, reproduces the main features of radio, X-ray, and γ-ray blazar surveys and also the extragalactic γ-ray background at energies ≳ 10 GeV. Here, we add a hadronic component for neutrino production and estimate the neutrino emission from BL Lacertae objects as a class, `calibrated' by fitting the spectral energy distributions of a preselected sample of such objects and their (putative) neutrino spectra. Unlike all previous papers on this topic, the neutrino background is then derived by summing up at a given energy the fluxes of each BL Lac in the simulation, all characterized by their own redshift, synchrotron peak energy, γ-ray flux, etc. Our main result is that BL Lacs as a class can explain the neutrino background seen by IceCube above ∼0.5 PeV while they only contribute ∼10 per cent at lower energies, leaving room to some other population(s)/physical mechanism. However, one cannot also exclude the possibility that individual BL Lacs still make a contribution at the ≈20 per cent level to the IceCube low-energy events. Our scenario makes specific predictions, which are testable in the next few years.