Agree about the tensile stresses... some of my first work as an engineer was related to failure of amalgams secured to teeth with steel pins. I met a dentist in the student union building at the University and after a little chat asked what he was studying. He was investigating the failure of amalgam fillings. I asked him what the material was like and he said it had a high compressive strength and a low tensile strength, very much like concrete. I asked him if it could be a tensile fatigue failure generated by the tensile stresses at the bearing of the amalgam and the steel pin. Did some FEM models using a 3D FEM program I had written and then we did additional studies using plastic and polarised light. Was fun...
Reinforcing is another issue for bearing. The ACI and CSA codes allow the increase in bearing stress just due to the confining nature of the loaded concrete with a small area loading a large area of concrete.
I'm not sure about ACI and CSA guidelines, although it seems obvious that if there is compression stress spread there is tension which should be carried by reinforcement.
Anyway, EC2 allows increase of up to 3 times of the design strength of concrete for crushing and spalling resistance (first potential failure mechanism) of the "partially" loaded area but the designer also needs to check the area for bursting (second potential failure mechanism), which should generally be done by strut-and-tie design of adequate reinforcement. If this reinforcement is provided and bursting is not the governing failure mechanism, then one can take advantage of the increased crushing/spalling resistance.
Not particularly allowed by EC, but "FIB Model Code 1990" (which is basically the basis of EC) allowed potentially increased bursting capacity for bulging compressive struts (which basically is this "partially" loaded area case where compression stresses spread out in a larger area) which are not adequately reinforced and the capacity (allowable bearing pressure) was influenced by the geometry of the bulging strut and the design tensile strength of the concrete. Of course this method could only be used for regions which where not expected to crack in the life of the structure because of any other influences/actions like shrinkage, etc. If cracking was expected and no bursting reinforcement provided then the bearing pressures allowed would be (by EC) around 53% of concrete compressive design strength.