I have always wondered about this. Why do Airbus side-sticks arithmetically add if they are never supposed to be operated together ? I think the intention is so that a TRE can grab the situation and pick up a wing in the event of a bad gust near the ground, or a missed flare, or a bounce, that a trainee does not react appropriately to, to avert a prang. If they also press the red button, fine, but even if they don't, they can still save the situation. I suspect that the "Dual Input" warning was provided for the trainee to be aware that their instructor was correcting on the non linked side-sticks.
Some years ago I was taxiing out at JFK and my Captain suddenly slammed on the brakes because I had slightly strayed off the centreline in a turn. The whole 230 tonne aircraft dipped and shuddered and lurched to a stop which was far too dramatic. If he had simply used his tiller to steer back onto the line and said something appropriate, it would have been far smoother and less alarming for all concerned, including the passengers.
.....Indeed as I myself did to another Captain when he went wrong - I just steered him back to the centreline with my tiller and asked if he was OK. No dramatic emergency stop required, just a gentle tweak.
Obviously, you absolutely do NOT want both pilots applying simultaneous inputs to their side-sticks routinely during normal flying or taxiing, but the facility is surely there for very occasional emergency corrections.