Since irrefutable proof of payment does not exist in the banking system, it is tempting to believe that there is no demand for it. This is not true.You're right—banks typically do not provide mathematically verifiable proof of payment in the cryptographic sense. Traditional financial systems rely on trusted intermediaries, such as banks, payment networks, and clearing houses. Payment confirmations (e.g., transaction receipts, bank statements, SWIFT messages) are based on institutional trust and record-keeping, not cryptographic proof.
In contrast, systems like Bitcoin or Ethereum allow cryptographic proof of payment through digital signatures and publicly verifiable ledgers. Anyone can independently verify that a transaction occurred without relying on a third party.
Irrefutable proof of payments allow, for example, for contracts in which the payee instructs to release the goods by the custodian to the payer upon verification of the irrefutable proof of payment. This is a very common contract situation.
The banking system is technologically hundreds of miles behind.
The main reason why they still exist, is because of regulatory capture that frantically tries to protect them from competition. Sooner or later, the banks will collapse anyway, and everybody who has deposits with them, will lose their money.