So if it's so obvious we 'are full' why are the Govt not looking to remove all the migrants (EU and non-EU) already here after we leave the EU?
Especially all the swathes of low skilled types working in our fields, factories, building sites etc.
That seems the simple solution to your obvious problem and would 'ease the pressure' and cure all the problems wouldn't it?
A good old fashioned sense of British decency and fair play would suggest that people who have migrated to the UK legally shouldn't then be made to leave except in exceptional circumstances (e.g. if they are convicted of a crime).