I'm sorry, but I fail to see anything in your argument that negates mine.
If you seriously want to cut immigration, you need a plan to achieve that end. At the moment, the issue is effectively left to the market, and what we have is the result of that.
The unemployed people who are 'already here' are rarely if ever fit to do the jobs we most need. Car washing and arse-wiping maybe. But what is your plan? Starve them to force them into these minimum wage jobs? You may just as easily starve them into crime. How many surgeons, doctors, nurses, dentists, engineers etc., do you imagine are on the dole? I suggest a handful at best and where they exist they have likely been struck off for valid reasons.
Increase the pay rate for arse-wiping to £18 an hour and you might, just might, get more natives to do the job. But then your Council Tax will go up, or your Income Tax, or both.
For the more 'professional' jobs we would need to train far more natives, and that will cost us. Again, up goes your Income Tax, or NI, or VAT. You probably will also need to increase the retirement age, as I said.
The problem with this country is that so many people will the ends, but not the means. In effect, they are in denial.