Why do Airbus sidesticks add their inputs? I do not work for Airbus and am not privy to their design documents, but here is what I reckon based on the over 50 real-time simulation models of aircraft flying controls I've written and tuned:
Summing the outputs makes most sense in avoiding discontinuities in position input, and provides the greatest flexibility if only one pilot at a time handles their side stick. A stick in centered position gives zero output. If a pilot transfers control to the other pilot, and no priority button is pressed, the flight control inputs from the two sticks have no discontinuity. Input from any pilot plus zero input from the other pilot works both ways, regardless of priority button status.
Only one pilot at a time should operate their side stick. If the non-operating pilot does provide a stick input, summing the two outputs results in a change in attitude that alerts the PF of the action of the other pilot. Averaging the inputs would do this to a much lesser amount, or not at all. Giving priority to left stick may be counterproductive: what if the captain has a heart attack while the FO is flying?
As long as proper procedure is followed and only one pilot at a time handles a stick, summing the outputs works best. However the situation breaks down when proper procedure is not followed: in times of emergency and urgency, in the absence of clear heads. Two pilots each continue to provide very different stick outputs - which one should the aircraft use? How can it make a decision of which of the two inputs is the valid one, if the humans have not instructed it? There is no fool proof solution to this, so picking the best solution when procedure is followed (summing the outputs) makes sense and provides flexibility and continuity.
There have been a couple of Airbus disasters where, in the confusion
leading up to the crash, both pilots have attempted to control the
aircraft with their sidesticks at the same time.
Consider the case of QZ8501. The captain is an ex military pilot, recognises the stall and provides proper recovery input to the stick. Had he instructed the FO to not touch the stick, he might have recovered successfully. But his ambiguous use of English (not his first language) caused the FO to follow perceived instructions and give a pull up input, of which the captain was unaware.
Summing the stick outputs did not save them, nor would have averaging the outputs. Transferring authority to the captain would have in this case, but now we would be waiting for Murphy's law: the captain panics while the FO gives proper inputs. I've heard anecdotal evidence of a case just like that, in gusty conditions at landing with an FO highly experienced in bad weather while the captain was from a warm and sunny country.
The basis of your question is a situation where proper procedure is not followed due to an emergency, and in this situation none of the options work very well. The only real solution here is to couple the sticks, and provide one output as a result of conflicting force inputs, like in physics. If there is no time for the rational brain to act, the only solution is to make use of the rapidly reacting sensory organs built into all humans.