> that the idea EU immigration was driving down wages was a cynical lie

It was a lie, EU citizens in the UK earn significantly more than the locals. If I wanted to be a member of the underclass, say a cleaner, I’d be in Germany, where I wouldn't have to pay 6K£ for pregnancy and 12K£pa to send a child to a half-decent primary school.

The thing is, from what I can tell the migrant workers doing these kinds of poorly-paid jobs didn't have to worry about the cost of pregnancy or sending kids to a half-decent primary school because their families weren't actually living here - they came over, worked, and sent money back home to their families in their home countries with lower cost of living. It's that which made it possible for them to work for less. You're right that if they and their families had to live in the UK permanently they'd need to be paid a lot more.

I am not saying that the set of poor Europeans living in the UK is empty, nor I’m saying that no grocer in the UK ever hired a Spaniard or a Frenchman for 11.1£ per hour while a local would have cost 11.6£ per hour, or whatever the poor earn in the UK, but that’s not a social phenomenon, it’s a guy, it’s at best an anecdote. Similarly EU citizens living under bridges to send remittances home, is not a social phenomenon in the UK, it’s a Brexit myth. You can check the data yourself. Again, you may know a guy that…, but that’s an anecdote. The mythological poor EU citizen that goes abroad to send money home and live in a rathole with 17 other poor souls, assuming it exist, is way better off going to Germany than to the UK. On a side note, remittances from Germany are 2.5 times those from the UK, from France 1.5.

EU citizens living in the UK on average earn a lot more than the locals and are better qualified. So saying that migration from the EU was driving down wages was indeed a lie.

But even if we assume that the UK had been swarmed by millions of illiterate cretins, there isn’t any evidence anywhere in the world that immigration suppresses wages permanently (there may be some temporary short term effects, but before, you say anything, the UK has been in the EU for decades, so we are indeed in the long run).