Which housing is over capacity due to immigrants? Immigrants tend not to have access to social housing. They can apply, but they won't get it ahead of anyone with a connection to the area. Immigrants do not put a strain on social housing - It's a myth.
The other myth is of course economic contribution. Immigrants put more in than they take out. We get the skills and labour without having to pay for the education and training. This is not to say that public services aren't under pressure but when you look at the bigger picture it is one of net contribution. Government policy not to invest in public services are why they are strained.
All housing is over capacity. That's why private rent is so high.
And you're just lying or misinformed if you don't think immigrants take social housing. I've seen it happen with whole neighbourhoods close to where I live. WHOLE neighbourhoods. We're talking close to a few hundred people all of one ethnic background.
Immigrants come to work but that simply drives down the cost for business of hiring people. That's how a market economy works. If there's more supply than demand prices lower. It's very good for the chattering classes since it lowers manual labour workmen as long as they don't do any profession that can be insourced but it's not good for working class people. It also means that the unemployment figures don't go down as they should.
That and as you said it increases pressure on public services which if the politicians don't then invest more it acts as a defacto cut because they get more tax but don't spend it appropriately on more public services.
So immigration is a way to 1. lower the costs for business and 2. make cuts in public services for existing citizens.
The fact it's a policy that Labour is a big believer in tells me how much the champagne socialists have fooled the working people for so long.