I clearly remember writing checks in the 1960’s when no account numbers were used. The accounts were held in the holder’s name. Signatures were checked to verify the account holder if there were any questions. Usually the bank teller knew the customers.
I guess the number of accounts and volume of checks processed were sufficiently low that name lookup could remain practical. Even without computers, I would think that if the volume of checks got very high, using account numbers would allow for somewhat faster information retrieval than using names. I'm not sure when various techniques of information processing were developed, though. Some optimizations which would be usable even for a paper-based account-management system may not have been developed until after computers came on scene, while others may have been used on paper before their use in computers. It's pretty amazing what people managed to do with paper when they had to.