The example I gave was based on a specific site with very dense granular soils near the surface & no groundwater table in the upper 20 ft. (so the analysis is only valid for that specific site). Having said that, yes, that pier can handle simultaneously the compression, tension, shear and moment loads mentioned earlier from a geotechnical view point. That is vertical settlement, lateral soil pressures & lateral deflection are acceptable. Obviously if you had 4 kip shear and 25 kip-ft, everything else being equal, the same pier would work.
When you do the reinforcement design, you can compute your Vu, Pu, Mu and then get your longitudinal & transverse reinforcement considering the P-M diagram. If we did your soils report and you had markedly different loading scheme then we will rerun the analysis and give you a new Vmax, Mmax, settlement, lateral deflection and lateral soil pressure. Then you factor those and do the reinforcement design.
There are situations when you have torsional moment, negative skin friction and large shear load all combined (Cantilever Sign or Metal Building Foundation for example). Then in those cases, if I am doing the foundation design, I check for Tu & Vu interaction, Pu & Mu interaction, after I satisfy the embedment depth and pier diameter is adequate from a soils point of view. It is iterative process. Sometimes anchor breakout may dictate the minimum drilled pier diameter required.
See attached references for more in depth study.