I wasn't talking about cash deposits. I was talking about paying bills. I can't go to my bank and give them the account numbers of all my bills and have the bank pay them for me. If I wanted to pay each bill in cash, I would have to schlep to each individual company I pay bills to and pay them.
For example, in Japan, you can pay bills by taking them to the local konbini - which is probably no more than a couple minutes walk away - then handing them the bills. They scan barcodes, take your cash, stamp the bills, give you your change. Done. No need for ID or a bank account.
The reason that people can’t easily pay bills in cash in the US is not because it can’t be done.
Banks do offer bill pay services, you just set it up and they mail a check every month. To your point, I don't think there's a way to get your bank to send the recipient cash every month, but I feel like that's a moot point since your cash is already in the banking system at that point.
Telephone banking used this before internet banking.