A "backup" ceases being useful when it serves as a distraction to the primary system or limits the primary system. Think for example a backup that needs to be done daily in a way that consumes an hour of an IT person's time, while also not being physically large enough to handle the primary system. That's the IT analogy here.
Paper strips may be computer fault proof, but they are also manual and limit the amount of air traffic a system can handle. Your choice is to keep it, or digitise it away. Running both systems in parallel as a backup serves not just to eliminate the benefit of digitising, but actively further reduces the handling capacity as you now have two systems to run in parallel.
Now there is a question of how much of an improvement digitising will bring. There's been good and bad examples of every digitising effort, and those which have worked well have resulted in increase in air traffic handling - e.g. London's air traffic control system was digitised 7 years ago to cope with increasing number of flights. It had teething issues, but overall worked well. France on the other hand had 2 failed rollouts of digitising CDG's air traffic control and are still on paper now. But that paper based system is now the bottleneck of the airport.