Are you certain that the EU is not directly responsible for some of the problems Britons complain about? I mean, I often hear a stereotype about Polish workers filling the menial job market in the UK and taking their earnings back to Poland where the cost of living is much lower and a pound goes much further, highly analogous to our situation with Mexico. I don't believe in supporting lazy/unproductive locals simply due to birthright, but if you in effect tell your poorer citizens "Hey, we're an international world now, if you can't work here move to Poland instead", obviously there are going to be issues. Once a generation is set in its ways, they don't like to leave. Another analogy would be California's prop 13; wealthy citizens all over the country move to booming California, drastically raising property values and therefore the taxes collected, and forcing older citizens out of their home since they could no longer afford taxes on their property. Californian locals voted in a referendum to give long-term property owners a significant tax break, increasing the barrier-to-entry from non-locals. Sure, it's anti-competitive, it suppresses the full extent to which an economy can grow, it's even xenophobic, but it's a perfectly natural reaction for people that don't want to change their livelihoods.
I won't say I'm certain; but then, I don't think you can be certain that the EU is directly responsible for problems in the UK.
As far as the stereotype you mention goes, it's more nuanced than that. Where Polish immigrants tend to fill the more menial jobs, as you say, is clustered in and around London.
Coincidentally, London overwhelmingly voted to remain in the UK, while the outlying districts voted to leave.
In short, it's people in the outlying districts who tend to fill those jobs, and Polish immigrants aren't taking those jobs from them. Meanwhile, Polish immigrants might very well be filling menial positions in and around London... but those people aren't complaining.
Now, as far as the effect this is having on the British economy... you can't blame this on the EU. This is Britain's own doing. Here's an excerpt from a
NYT article in 2007:
When Poland and nine other new members, most of them former Communist countries, were admitted to the European Union, many West Europeans feared an influx of cheap labor. In May 2005 in France, opponents of a new European constitution used the labor threat — personified by an archetypal “Polish plumber” who would steal French jobs — to help defeat the proposed constitution in a national referendum.
But Britain, along with Ireland and Sweden, welcomed workers from the new European Union members — partly because they took physically demanding, minimum-wage jobs that many native-born Britons snubbed and partly because a wide range of industries in this country were suffering labor shortages.
Today, the reputation of Polish construction workers, nannies and caregivers is so high that other East Europeans sometimes say they are Polish to increase their chances of being hired. At Strathaird Salmon, a fish farm in Scotland, more than a third of the employees are from Poland.
British workers didn't want those jobs, and Polish workers took them - primarily comprising a demographic in and around London. Outside of the major metropolitan areas, British citizens purportedly still took those jobs, yet they are the ones complaining.
The moral of the story is this: those Brits that are suffering economically outside the metropolitan areas are projecting their problems onto the influx of immigrants, when other systemic issues are very likely at play.
And
that, my friends, is why xenophobia is absolutely a centrally motivating factor in the Brexit campaign.
I dunno, wealthy EU members generally don't go to war with other countries developed either (with obviously some small exceptions like the Falklands war),
for good reason. Particularly keep in mind the countries that were members up through the 90s; aside from problems with the IRA between the UK and Ireland, they weren't nations that particularly had any reason to kill each other (and it might be worth noting that IRA terrorism continued for decades after joining the EU regardless). What conflicts do you think it prevented?
I won't speculate. I'm simply saying that it makes sense to me that the EU exists in the first place.