A thermodynamically consistent continuum theory for single-species, step-flow epitaxy that extends the classical Burton-Cabrera-Frank (BCF) framework is derived from basic considerations. In particular, an expression for the step chemical potential is obtained that contains two energetic contributions---one from the adjacent terraces in the form of the jump in the adatom grand canonical potential and the other from the monolayer of crystallized adatoms that underlies the upper terrace in the form of the nominal bulk chemical potential---thus generalizing the classical Gibbs-Thomson relation to the dynamic, dissipative setting of step-flow growth. The linear stability analysis of the resulting quasistatic free-boundary problem for an infinite train of equidistant rectilinear steps yields explicit---i.e., analytical---criteria for the onset of step bunching in terms of the basic physical and geometric parameters of the theory. It is found that, in contrast with the predictions of the classical BCF model, both in the absence as well as in the presence of desorption, a growth regime exists for which step bunching occurs, except possibly in the dilute limit where the train is always stable to step bunching. In the present framework, the onset of one-dimensional instabilities is directly attributed to the energetic influence on the migrating steps of the adjacent terraces. Hence the theory provides a ``minimalist'' alternative to existing theories of step bunching and should be relevant to, e.g., molecular beam epitaxy of GaAs where the equilibrium adatom density is shown by Tersoff, Johnson, and Orr [Phys. Rev. B 78, 282 (1997)] to be extremely high.