As many small business do, one has to maintain a record of invoices issued and payments made. In effect it’s a debtors or invoice issued ledger, ie it records who owes money and who has paid. But this ledger, or record, does not fully integrate into the accounting system because the accounting system doesn’t care about non-cash economic movements – as you point out.
Of course where the two systems do meet is when payment is received. The “off-line” debtors ledger records a payment, while the double-entry cash accounting system debits bank/cash and credits sales.