This already happens. I have not been trained to fly an airliner, but I believe that the priority is something like stall, gpws (flying into the ground), tcas (flying into other traffic), master warning, and master caution. These all have their own lights and sounds in the cockpit, and if I know them, so does the flight crew. Presumably the 150 messages that needed to be looked at were all "caution" messages. Too many messages, yes, but nothing that endangered the safety of the flight.

In the AF447 case, the crew was getting the stall warning. They did the exact opposite of what you are supposed to do to recover from the stall. Computers were not the problem. (The same thing happened to that Colgan Air flight a few years ago. The crew reacted to the stall warning by stalling the plane even more. The plane tried to physically force the controls to nose down, which is how you recover from a stall, but the pilots fought it. Bad UI was not the problem. Bad flying was.)