Of course it is, but I can understand that if you haven't got a house you might believe the myth
Which is, of course, a failure that rests squarely with the UK government, and has absolutely nothing whatsoever to do with the EU.
Presumably the Brexiters have a plan to tackle the Immigration Tribunals which over-rule Home Office decisions to refuse entry or deport people? The Brexit argument is that Tribunals are applying EU law and if freed from this requirement, the Home Office would have more control of who it lets in.
Since there is a labour shortage, an Australian style points system favoured by Brexiters (Australia apparently has twice the number of immigrants per head as Britain) would still allow immigrants in to put pressure on infrastructure such as schools, housing and healthcare. More than 260,000 immigrants arrive in Australia each year on temporary visas - employers can sponsor foreign workers to come in on temporary visas.
http://www.theguardian.com/uk-news/2016/jun/01/brexiters-missing-point-australian-style-immigration-conservativesAlso, I haven't seen any evidence that Immigration tribunals, which are independent of the government, would rule in favour of Home Office decisions if we were not in the EU, or how this will solve the problems of people on low incomes.
I don't see how more controlled immigration will make renting or buying in England affordable - currently on average people spend half their pay on rent with it going up to 72% in London.
http://www.theguardian.com/money/2015/jul/16/tenants-in-england-spend-half-their-pay-on-rentPeople who are unemployed or on low wages or zero hours contracts can't afford rent and landlords do not want to rent to people on Housing Benefit as they have an image of being bad tenants - so those problems won't go away by controlling immigration.
I also hope a Brexiter could explain how controlling immigration would do away with zero hours contracts, or stop a business bringing in immigrants as cheap labour for low-skilled jobs because there are jobs that need to be filled and local people will not work for the wages offered, either because they are better off not working based on current welfare policies or because they don't have the skills. Given the high cost of living, especially rent, businesses can't afford to pay the wages local people want to be paid for low-skilled jobs.
Due to the lack of housing, even people in relatively well-paid skilled jobs often feel like they are struggling to afford rent and the high cost of living, and it doesn't get any easier for them given that in today's age of technology the solution for businesses to high salary costs which they can't recoup in fee income, is simply to outsource many tasks off-shore, especially in relation to many office jobs. One solution I suppose is for everyone but the wealthy elite to downgrade their expectations, aspirations and retirement prospects.
The govt issued a press release a while back claiming they were working with the Met police to clear off homeless migrants, who were in the UK illegally since they had no means to support themselves, and put them on planes or coaches back to say Romania or Bulgaria. But the numbers sent back are not increasing public trust in the system, even though the Met police started checking the immigration status of people they arrested for other crimes and apparently found only a very small proportion of people arrested had immigration issues. Also nothing seems to stop deported EU migrants from returning. On the other hand the number of homeless Polish sleeping on the streets has dropped as the Polish economy has improved so apparently a lot of homeless Poles went home. If Britain leaves the EU it seems likely that this will adversely affect the economies of other countries in the EU and make Britain more appealing.
And here perhaps is the nub of the issue - whether we should prioritise people who happen to be born in the UK whether or not they have contributed to the UK or whether we should priories those who have contributed to the UK wherever they might have been born.
If you do not have a home or are unemployed, regardless of your contributions to the system through NI or tax, people have this expectation that they cannot be left to starve or sleep on the streets. So people in priority need - pregnant women or people with dependent children or old or mentally ill or disabled or young people or those leaving the armed forces or prison - will always have priority over those who have paid into the system but do not fall into these categories.
One of the issues for those focusing on immigration is that migrants with dependent children will get help with rent, when they should be sent back after 3 months because they don't have a means of supporting themselves and are therefore not meeting the criteria for having a legal right to be in Britain. But again this is a tiny part of the huge housing problem, high cost of living, soaring healthcare costs due to high treatment costs or an ageing population etc in England so not sure how tightening immigration will solve these problems and I am not clear how Brexiters plan on doing a better job of immigration enforcement, given all the cuts to spending.