The recently identified possibility of ground-state cooling of a mechanical oscillator in the unresolved sideband regime by combination of the dissipative and dispersive optomechanical coupling under the red sideband excitation [Phys. Rev. A 88, 023850 (2013)] is currently viewed as a remarkable finding. We present a comprehensive analysis of this protocol, which reveals its very high sensitivity to small imperfections such as an additional dissipation, the inaccuracy of the optimized experimental settings, and the inaccuracy of the theoretical framework adopted. The impact of these imperfections on the cooling limit is quantitatively assessed. A very strong effect on the cooling limit is found from the internal cavity decay rate which, even being small compared with the detection rate, may drastically push that limit up, questioning the possibility of the ground-state cooling. Specifically, the internal loss can only be neglected if the ratio of the internal decay rate to the detection rate is much smaller than the ratio of the cooling limit predicted by the protocol to the common dispersive-coupling assisted sideband cooling limit. Moreover, we establish that the condition of applicability of theory of that protocol is the requirement that the latter ratio is much smaller than one. A detailed comparison of the cooling protocol in question with the dispersive-coupling-assisted protocols which use the red sideband excitation or feedback is presented.