The real question is, how did we do this before the age of computers? Paper, pencil and human experience ... all easily replaced by a computer ... until something goes wrong.
I believe airlines had employees who did basically nothing but make these calculations.
The thing is, they were fallible too. A surprising number of old accident investigations found that the weight-and-balance calculations were incorrect, although in the vast majority of cases they were close enough that it didn't create a problem. A computer system that reliably gets it right but occasionally goes down entirely is probably better than a human who occasionally sends a flight into the air based on incorrect numbers, although neither is ideal.