ID cards do nothing to limit immigration from the EU. All they do is allow you to track the numbers better (at cost to the UK government / taxpayer). You talk about leave disinformation (and yes there is some of it) but the story you've posted is remain disinformation.
You can see if someone outstayed the 90 days. You can stop people claiming benefits when they are not entitled to claim them after 90 days. You can control Minimum wage more strictly to stop wage undercutting. I cannot understand objections to providing id to claim benefits. If I wanted to get money from a bank account, I would have to have a bank card. We are the only country in the bloc without ID cards and we moan about benefit abuse and think EU citizens are claiming things they aren’t entitled to. So, wreck the economy instead of introducing ID Cards.