Earlier quoted context omitted.
As someone who's just got their new British passport, faking the 3 pictures on it, and the whole passport itself, does not look trivial at all... I think it is much, much, much more common to have dodgy employers/landlords who do not carry out the checks at all because they are fine exploiting illegal immigrants, and no type of ID card would solve that...
Most people are paid cash in hand if they're working illegally realistically. I'm not sure that would change. But in enforcement it might since you could theoretically make it a legal requirement to produce the ID, that's the norm in many other countries.
E.g. if a firm is doing better than its peer group with less employees on the books, something is suspect.