Yes, and to be paid via PAYE you need a NI number.

The prove right to work is a slightly newer thing thats additional

The nuance is that you can have a NI number, then have your visa lapse for whatever reason - you still have the NI number. Hence the requirement to prove your right to work through another means.

Previously you could use proof of British nationality or a physical biometric residence card - but they've been replaced by the digital share code system (which tbh hasn't been too bad)

No, those are still the ways of proving you have the right to work, it’s only if you’re not a national that you need the share code.

Sorry I worded that poorly - I was trying to make the point that citizens prove their right to work using passport/birth certificate, and until recently visa holders used a physical BRP, and now a digital system (which oddly enough uses your expired/redundant BRP number as a username)

The share code stuff is not for nationals. It’s not clear to me exactly how it works and whether it’s scalable.