We present further development of a pulsar emission model based on multiple streams diverging away from the magnetic dipole axis, and forming azimuthally-structured fan-shaped beams. It is shown that this geometry, successfully tested on profiles with bifurcated features, naturally solves several classical pulsar problems and avoids some difficulties of the traditional nested cone/core model. This is best visible for profiles with several components, such as those of class T, Q and M, because they most clearly exhibit a range of effects previously interpreted within the conal model. In particular, with no reference to the flaring boundary of the polar magnetic flux tube, the stream model explains the apparent radius-to-frequency mapping (RFM), including its reduced strength for the inner pair of components. The lag of the central component (apparent 'core') with respect to the centroids of the flanking ('conal') components can also be naturally explained with no reference to emission rings located at disparate altitudes. The stream model also reveals why the millisecond pulsars, despite their more strongly flaring magnetic field lines, do not exhibit as strong RFM as the normal pulsars. The model is then successful in reproducing properties of so disparate objects as the M-class and millisecond pulsars, including some peculiarities of the latter. With no hesitation we, therefore, advance the view that pulsars have fan beams generated by outflowing streams, whereas the nested cone/core beams may well not exist at all.