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No, that's the thing. You and others consistently say "no one wants 'open borders'" but baulk at any policies designed to restrict immigration, ask 'what would you restrict?' or dismiss it as a secondary issue altogether.
I remember bullet pointing areas where immigration directly influences the cost of living and instead of responding, you reacted with '

'. That is fine but this what the political right is picking up on and the political left is not and they're being punished for it outside of the urban middle-classes.
I, and I'm sure many like me, would much prefer it if the jobs were taken by local people.
As you pointed out with HGV drivers and the supply/demand during covid, we've got a HUGE undersupply in the care sector, so wages should be going up to reflect this, correct? Wages go up, more people are willing to do the job, supply and demand even out.
Only that's where we reach the problem isn't it? Because increasing those wages will require large increases in spending, and who's paying for that? Talk about increasing tax? "We're already taxed too much!" Tax the wealthy? "They just leave and you end up receiving less tax!" Borrow more? "We have to balance the books!" Print more money? "You'll devalue the currency!"
So ultimately nothing happens and we can't afford to pay those extra wages to get locals to do it. But those positions still urgently need to be filled, and then the only short-term option left is to bring in workers from abroad who will work for low money. Which itself has huge consequences in the short, medium and long term with regards housing, public services, pensions and welfare etc.
The left are suggesting ways to reduce the need for foreign workers by suggesting a more equal sharing of the pot to enable the higher wages to get locals to take the jobs instead. It's the right that are baulking at policies that would restrict immigration by not requiring so many of them needing to come here to work.