//Are you suggesting my colleague was wrong – and it was not the fault of the EU he could not get the school of his choice for his children?//
You need to rationalise the argument a bit more closely.
As a condition of our EU membership, any of around 450m people from across the bloc had the unconditional right to settle and work here and, as a consequence, have their children educated at UK State schools. Local Authorities do their best to forecast demand for the schools under their control and provide the forecast number of places. This is a bit difficult when there is the potential for so much uncontrolled influx.
To give you some idea of the problems it presents, when the eastern expansion of the EU took place in 2004 the percentage of the UK population who were born in the EU was about 1.5%. The UK decided not to take advantage of the seven year moratorium on migration from those countries and between 2004 and 2016 that percentage rose to over 6%. Upon the 2004 expansion it was estimated that migration from those countries would be "no more than 10,000" per annum. In the first three years the total figure was 230,000. By 2014 the total was more than 800,000. So it was little wonder that Local Authorities struggle to provide sufficient school places.
So your colleague is wrong - it was not the fault of the EU that he could not get the school of his choice for his children. It was, in fact, the fault of the UK government for laying the country open to such a ridiculous scheme which allows such uncontrolled and unconditional rights of settlement to so many people. But he was right to cast his vote the way he did because the only way to end that preposterous situation was for the UK to leave the EU. In short, his actions were right but his reasoning was wrong.
The population is wrong when it "blames" the EU for the things it does that they don't like. They had no control over what the EU does (which is one of the things that many people don't like). The blame lies with successive UK governments for allowing the country's affairs to be increasingly removed from their remit.
The seeds of Brexit were sown in 1992 when that nice Mr Major signed the Maastricht Treaty - the first significant move towards a federal Europe. They were nurtured through successive increments towards that goal and our membership's fate was effectively sealed when Gordon Brown signed the Lisbon Treaty (originally entitled the "EU Constitution") in 2009. Mr Cameron tried and failed to recover the electorate's confidence in the EU. He asked for next to nothing in the way of "reform" and came back with slightly less than that. Your friend may not have known it, but he was simply doing the only thing he could - the only thing any of us have been given the opportunity to do - to end the nauseous farce that was our EU membership.