What is reasoning behind the common wisdom of limiting the temperature rise of a PCB trace to 5/10/20°C?
Many things go into designing the width of the PCB trace, including temperature rise for current. Others are voltage drop, impedance, PCB fab capability, cost, packing density.

However, temperature rise is rightly one of the 'do not exceed' specifications. 

A rule of thumb is just that, something you should follow most of the time. You will always be able to find edge cases where are higher rise is permitted, if you do careful calculations. 

Part of the benefit of a rule of thumb is that if you follow it, your calculations don't have to be too careful, there's already a big margin of error built into the rule.

A peculiarity of temperature rise is that it's proportional to current squared, not just to the current. This reduces the importance of choosing one specific value. The current that gives a 20C rise is not twice the 10C rise current, it's only 1.4x the 10C rise current. If we double the 10C rise current, we get a 40C rise, which is starting to feel uncomfortably warm.

Why run a board cool? All sorts of good reasons. Component cooling requires a low ambient. Component life drops very quickly as temperature rises. Margin for operating in warm places (inside a car cabin in bright sunlight) is good. Debugging, run your finger over the circuit to find toasty components, you'll get confused by hot traces.

There is no one killer reason to run a board cool, and no one reason to choose 10C rise versus 20C rise. However, few designers feel inhibited by following this 'rule'. It's rarely the thing that sets the limit. If we do find ourselves in some corner case where the specification cannot be achieved by sticking to some arbitrary temperature rise figure, then we calculate and test the heck out of everything, to see what effect on lifetime and cooling higher temperatures will cause.