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Previously you could use proof of British nationality or a physical biometric residence card - but they've been replaced by the digital share code system (which tbh hasn't been too bad)
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 stated reason is to stop illegals working.
Unfortunately we have an ID for working, called a national insurance number. We literally can't get legally paid without it.
So a National ID card ... Is irrelevant. You still need this number for benefits, etc.
I've got an NI number, a driving license and a passport. Not to mention a NHS number.
I don't need another form of identification to link together everything about me so my government can leak everywhere.