I agree that there has been a failure by successive governments to use current controls on non-EU immigration. The worst problem is the illegals running at 150,000 per year, who come in and use resources, live in the black economy and don't pay taxes. Most have come in legally, their visa expires and they disappear into the population. There are strong grounds for believing that council tenants in the Grenfell Tower were subletting to illegals, and thus some of the unfortunates that perished will never be known. There was a good article on this in the Sunday Times yesterday by Dominic Lawson. It was in fact Labour's Charles Clarke who tried to tackle illegal immigration around 2004 by seeking to introduce a national biometric ID card scheme - when Theresa May became Home Secretary in 2010 she abandoned the idea, which for me is a black mark against her, and the government needs to look again at ID cards.
As to legal immigration, we need control because with high levels of immigration running as they are and thus the strains on public services etc, the problem will only get worse. After Brexit, we invite in who we need - we don't give rights of entry to those we don't need.