I mean, the premise is more or less the same as that of keeping your money in gold bars under your bed, except without the physical labor. So yeah, seems like a bad idea.
Yes, in theory you could have done everything with checks. It didn't completely happen though, until after computerisation. In the UK, it turns out that employees had the right to receive pay in cash as late as 1986! And taking it away was controversial.So it may be that you are right that computerisation wasn't essential, but most likely the reduction in cost of banking spread it to enough people that armed robbery became less of a risk.