I agree with PPs that we are not over populated in regards to space. But we are over populated in regards to quality of life and public services.
Say we let in as many migrants as want to come. These will mostly be low paid workers, so likey to take more out than put in to the tax system.
We then actually build the homes and public services needed for everyone. But because we have a nation of low paid workers - there isn't as much tax to pay for all that stuff. So the quality of schools, health care, police etc etc is low, lower than it already is.
Housing is cramped, what were large family homes become HMO. People build on top of existing homes(already happening in London). Affordable housing becomes cramped flats instead of family homes.
We limit the outdoor space we have, what we do have becomes cramped and overly busy.
Roads and transport are over used.
We end up in a country that yes, gives housing and low level public services to everyone, but everyone's quality of life goes down.
Things are pretty good here right now. Other than in cities most families can expect a house with a garden. We have adequate public services. We have outdoor spaces that everyone can enjoy, farm land to produce our own food and although badly managed and distributed - enough public money to keep the whole country going and in a decent standard.
Slowly we are seeing that standard of living slip. Personally I don't see how cramming a load of low income workers into our country is going to help that. If we want to retain this standard of living for our children, or hopefully improve on it, we need to be a bit more selective about who we let in. That might be selfish, but is it really our job to provide a home to every inhabitant of a shit country, just because ours is nicer?