For a start, I'd like to make clear that by being blunt about immigration's role in the late decline in UK public services, I'm not blaming the immigrants themselves. If they couldn't enter the UK, immigrants would have had many alternative destinations like in Europe or the US. Like you, I agree that what matters is citizenship, rather than nationality.
Anyway, the EU does have its draw backs. There are plenty of examples of where it has been corrupted by industry lobbyists in order to protect themselves from new innovations (but what country where policy, for hundreds of millions of people and an $18 trillion economy, is made in a single city isn't?), and the EU is not that innovation-friendly in general. EU regulation is the biggest reason as to why Europe lacks big tech companies in comparison to the rest of the world. It's almost shameful, really. It's also failed on its Common Agricultural Policy (which is one of the EU's primary functions), which has very good use but due to the intricacies that weren't quite thought through farmers receive more money to store excess harvest (for years with bad harvests) and they'll all go on strike if the EU tries to bring the payment down to the right level. Regardless, the UK pays about 1% of its GDP to stay in the EU, and I'd say that's a bargain price for all the benefits brought with it - peace, investment, students, labour, jobs.
Aside from that, though, I'd say that it's a wonderful thing. People forget that Europe has been one of the most turbulent regions in history and to somehow manage to unite all the different nationalities and cultures in peaceful unity is something we take for granted. Without that kind of unity, the continent divides very, very quickly. With the EU, the Single Market provides pressures strong enough to keep countries in check, therefore preventing the kind of domino effects that would lead to conflict. Freedom of movement stifles corruption and bad governance as citizens can emigrate with relative ease. This does have the opposite effect of draining less developed nations of there skilled workers, yet the EU spends about 40% of its budget on development through its schemes like Horizon 2020.
The EU is far from perfect, but to leave it is a huge mistake. It just needs reform. The immigration policy is perhaps the EU's most hated aspect which it'd do well to fix in order to quell the far-right movements that are tearing it apart. Yet, the root of the problem lies with stagnant incomes, austerity, and bad economic thinking. Don't forget that one of the most influential economics studies released in favour of austerity was found false due to the economists missing out a line of date in their spreadsheet (but by then it was too late).
Austerity would not have been necessary if the Conservatives weren't so reluctant to raise income taxes on the rich and introduce a wealth tax. Inequality is a proven fact and it gets exponentially worse the longer it goes without treatment - like a tumour. There is some evidence to suggest that soaring inequality and a broken social contract are all down to globalisation - and therefore worsened by the EU. Exposed to international competition from the '70s onwards, Britain's manufacturing base has quickly died off or been bought. Shareholders force management to maximise profits in order to stay profitable (otherwise they'll sell and invest elsewhere), leaving employees with wage freezes or cuts, employee benefits being stripped away, full-time workers being rehired at lower wages as part-time employees. The list is staggering, and convincing, though I haven't checked the data.
If I've gone from talking about immigration to shareholders, I've probably gone on a rant. I'll shut up now.