I print each days itinerary and bring the itinerary for that day into the park with me. I do it mostly to be able to see where everyone is and needs to be when the app is flaking out. I have not had to use it for problems with ADRs. Carrying a piece of paper isn't hard and if it makes you feel better - do it. Just realize that if their computer disagrees with your paper, they do not have to honor your paper. You could book things, print out the paper and then cancel them to avoid charges if you don't show up. In that case you would have the paper, but no actual reservation.
In rock paper scissors I would bet that computer takes the place of scissors and beats paper.