A lot of people are scared by the concept of 'freedom of movement' but these same folk hypocritically enjoy the freedom to pop into Yorkshire, Derbyshire or even Wales whenever they feel like it. The FOM in the EU is actually less free than that*, but it's really only the same as having bigger borders. It is not the same as having 'open borders'.
The Schengen countries exchange all sorts of data to which we are no longer privy due to Brexit, so we actually have less information about people who come here, legally or otherwise, than we did!
The most effective tool for controlling illegal migration would be to have ID cards and also to completely ban the cash-in-hand economy. But the odd thing is, the people most exercised by the topic are also the ones most opposed to these practical measures.
*You can skive on the dole in the UK county of your choice, but the likes of Germany limit non-native EU nationals to three months. We could never be bother to enforce this, as it was easier to blame the EU.