The problem is, increasingly over the past 20 years, we haven't wanted a 'working class'. We treat low-wage jobs with increasing distain. Everybody's kids have to go to uni so they don't end up working in factories/cafes/as carers.
The post-Thatcherite attitude is you have failed if you don't have a profession. So we now have at least two generations of children who think such work is beneath them, because that's what they have been told. In addition, it's become impossible to live on the lowest wages.
White British kids don't want the work. Why would they, when the pride in doing an honest day's work has been slowly stripped away? When you can't even afford a bedsit on the wages without working silly hours?
Immigrants haven't taken our jobs. We have given them away through a culture of expectation. Expectation that jobs people once did their whole lives should now only be stop-gaps/student work. Expectation that we should all be striving to earn above the minimum, driven increasingly by the fact we can't live on the minimum.
Leaving aside the cost of general living, there's a far greater expectation now that we should all live to what would once have been a middle-class standard. Holidays every year, etc.
Whilst people might not like the migration, at the same time they don't want to fill the gaps migration is filling at the lower level. They don't even want to fill in the gaps at the higher level, such as doctors and dentists, because if you can do those things, you are capable of doing other things which incur less debt at uni and pay more.
As a society, we have stopped valuing certain jobs/professions. Whilst it has increasingly become about COL, the attitude started much before this.
I say this as someone who has always been a low-earner, who started working when it was possible for me, a factory floor worker at 18, to rent a flat on my own.
We have sold our industry. We have got stuck in the model of 'betterment' to the point where our kids would rather sit on their arses until they get the career they now expect.