Just so we are clear. The UK took like 20K refugees from Syria. They were beaten by Sweden, Switzerland, Sudan, Spain, Greece, Egypt, Norway, Netherlands, Jordan, Denmark, Belgium, Austria, Lebanon, Armenia and Canada. EVEN FUCKING IRAQ. Hell, Australia took 15K and they aren't in the EU. The UK paid a bunch of money so they could be in this privilege position of not taking refugees. So, they could have paid less if they wanted, but they chose not to. And I feel like I have to repeat this: Australia, who has just a bit more than a third of the UK population, took almost the same amount of refugees. But the UK was 'doing its fair share.' Yes, we know, you wanted to do nothing. We get it
Now, you could complain that the EU shouldn't have taken too many refugees. They didn't. They took less than a quarter 1/5 of all Syrian refugees. Except Turkey, who took 4 times the amount of the rest of Europe. So, IMO, the EU didn't do its fair share.
As to your claim of picking and choosing. I don't know what clearer evidence of picking and choosing anyone could provide, because you pretty much proved the EU point. So freighted that a bad 'economic immigrant' might slip through, the UK decided they would make their own rules. See also: This utter nonsense by Pretti Patel and her jet ski solutions.