I imagine the number of errors/deaths/kerfuffles due to the opportunity cost of professional time, not to mention transcription and human errors of such systems are not inconsequential...even if they aren't at the forefront of our minds, and even if some of them haven't been measured accurately.
This means that you have basically added another data entry level, increasing both time and error rate.
The goal is not to remove humans from the equation, since that is probably both impossible and undesired, but to separate humans and computers into supporting each other in the task at hand by specialising into what they do best: computers for tasks that can be broken down into repetition, replication, automation, speed, validation, and volume, and humans for creation, ambiguity, complexity, context, service delivery and interpretation.
At the moment i'm guessing there's a fair bit of humans doing the jobs computers are good at in hospitals around the world...
The big 'but' is that there are so many hidden requirements here. Privacy is a huge concern. Data on patients may not leave the hospital. With the devices we have these days, it is almost impossible to create a near offline system. Almost all devices are connected to "the cloud" in some form or another, making them vulnerable.
The fact that health insurers are grossly incompetent in this aspect does not mean that hospitals should go down to that level.
Another hidden requirement is that this system has to work, always, no matter what happens. Paper notes may not be efficient, or complete. However, you can be pretty sure that while the hospital is still standing, this system works. Making a digital system that is as reliable is a very hard task.
All this will make such a system a huge investment, not only the system itself, but also in training. Balanced against the cost, it is not so evident that this is really an acceptable investment.