I really don't see what the issue with ID cards is
They would cost a massive amount to implement and maintain.
In order to "make life difficult for fraudsters, criminals and people wishing to stay in the UK illegally." ID cards would need to be demanded often - by doctors, schools, nurseries, the Police etc etc I regard this as an interference in my daily life I can do without. Far better to actually deal with the people you mention, than inconvenience everyone.
Considering how useless we are at dealing with illegal immigrants now - do you really see ID cards helping?
In order to be effective, every single person would need a card - what about travellers, homeless people, Old people in nursing homes who have dementia?
A lot of other countries that have ID cards don't really make much use of them so they are a bit pointless - but they can trigger unrest, like the Paris Banlieu riots where Police continually demanding ID from everyone was being used as harrasment.
Advocates of ID cards haven't considered the practicalities properly and haven't thought about how admin typically works in the UK.
There are a lot more objections - but that will do for starters - they are a bad idea, a solution looking for a problem that doesn't exist.