Interbank payment systems are actually provided by central banks or the government (or affiliated organizations of either) in quite a few cases.
Europe has TIPS (an ECB-operated implementation of the SEPA Instant Credit Transfer scheme), India has UPI (which is pretty close to the central bank, as far as I understand) etc.
In the case of FedNow and TIPS, there are private alternatives as well, such as RTP in the US or EBA Clearing's SEPA Instant implementation in Europe. This is similar to ACH – there's both a public (FedACH) and a private (The Clearing House) implementation/network available.