The saddest thing about Brexit, is the rhetoric used by the Conservative party was complete guff. They never used the tools given to them to curb immigration, because they didn't want to.
Tool 1 - Changes to taxation.
I've mentioned this many times on here, but simply changing taxation could have cut immigration fairly drastically. They actually made it worse, by increasing the 'take home pay' from low paying jobs (not that I am against that).
https://infacts.org/government-use-taxation-restrict-immigration/
Now there are sensible reasons not to have implemented this, you can question whether it's fair, whether it will increase poverty, how it will affect returning ex-pats and why the government should have to resort to such backwards taxation anyway... but it was a tool that no government chose to use to curb immigration.
And actually, when the worst part if EU immigration is the reduction in low-skilled wages over time, this would have reversed that.
Tool 2 - Ask migrants without a job for three months to leave.
David Cameron claimed he'd got a concession that allowed migrants that haven't had a job here for 6 months to be asked to leave. But we already had that right.
And we (Theresa May/David Cameron) just didn't use it.
Tool 3 - Ask migrants to register with the local government,
Pretty much every other country requires you to register with the local government.
Tool 4 - Ask everyone to get Health Insurance (UK Citz would use the NHS)
A bit of a weird one. Healthcare is entirely devolved to the countries in question. This goes against the whole point of the NHS, but...
You are allowed to require people working in the UK to have private Health Insurance. You could also give everyone who has been in the UK for over a year "NHS Health Insurance." EU migrants would still get this NHS Health Insurance after a year, but would need to get their own Health Insurance to start with...
There are any number of tools available. And yet, the Conservative government didn't choose to use any of them. Why? Because Immigration is good for the UK.
It's all bollocks.