The WF/WH-1000XM4 are not gaming oriented headsets. If you want to game on the PS5 and have a Sony headset, the In-Zone headsets offer wireless connectivity for the PS5. And no, they do not use normal bluetooth, but a proprietary 2.4 Ghz connection as mentioned in a comment above.
QuoteI see no reason why other Sony headphones can't do that with a simple firmware update unless they lack the requisite hardware.
that is because the current headsets do lack the hardware required for low latency gaming and a firmware update won't add <30ms delay capabilities with current tech. Perhaps LE Audio, but we've yet to see real world application of the tech in a production consumer headset.