Just make sure you have the passenger numbers to start with. Its known as plan b
Weight & balance could indeed all be calculated by the pilots or some simple software, but it's a bit more complex than that. The load sheet is used by the ground handlers to know what items to put where - which is determined by things other than just weight. Aircraft carry freight as well as passengers and baggage, and not only would the pilot need to have a list of all the freight & weight thereof (which he would not get if the computer is down), but will also need to know the volume and shape of that freight and any special requirements. Putting 5 items of freight weighing 2500kg into the forward hold may be great for the weight & balance, but the items might not all physically fit into that hold.
Then there's a multitude of restrictions that the aircraft captain does not necessarily know about. The MRI scanner mustn't be placed in the forward hold because its magnetic field could upset the navigation systems. Certain types of live animals must not be put in the same hold together (dogs and snakes for example). There are lots of chemical combinations that must not be transported in the same hold. Fruit & veg should not be placed in proximity to fresh fish. Plus 1001 other rules that the pilot would not be expected to know but the loadmaster must take into account when preparing the load sheet.