Re: "How would it have protected them?"
"So, in the Windrush example, they would have been registered in the ID system soon after their arrival, and they would have not depended on unrelated record to demonstrate their status as legitimate UK citizens."
Well no it wouldn't because whey they arrived they were not uk citizens and unless they've applied for UK citizenship they're still not UK citizens. They arrived as Commonwealth citizens which gave them certain rights to remain. Depending on where they came from they lost that citizenship when the countries they came from got their independence and became citizens of that country. Their right to remain then hung on the fact that they had it when the arrived and they had not lost it in the meantime (by leaving the country for more than 3 months for example) so an ID card would show when they arrived In the UK but not that they had been continuously resident for the intervening period. A dirving license issued the same year would provide the same proof of arrival.
"The death certificate was already submitted - I used my ID card to obtain it at the city office - should they issue it to the any person who comes up to ask for it?"
Well in the UK if you want a copy of a birth, death or marriage certificate you can go to any Registrar and buy one for about £30, see https://www.nrscotland.gov.uk/registration/how-to-order-an-official-extract-from-the-registers, or you can get them online at https://www.scotlandspeople.gov.uk/r for ~£2. If you want a copy of my dads his name was James William Rae he was 80 and he died in Lanark. What you're going to do with it I don't know, all it does is prove that he's dead.