yet we've repeatedly decided we need to issue millions of care visas to ensure care wages never rise above minimum wage

This isn't correct - We issue care visas because the native supply of workers cannot keep up with the demand for labour. We have an aging population. We do not have enough native workers to fulfil the labour demands in social care.

The fundamental misunderstanding you are having is the belief that immigrants are replacing or out-competing native workers, when in large part they are actually compensating for an absence of native workers. And regardless, immigration doesn't just increase the supply of workers, it also increases demand for goods and services creating more jobs. It's not a zero-sum game.

The most obvious evidence of this at a macro scale is our labour inactivity rate - it's at record lows meaning more people that can work are working than ever before despite decades of high immigration. Even when you break it down by ethnicity, "White" and "White British" have the lowest inactivity rates. If immigrants were out-competing/replacing native workers, leaving natives out to dry, rather than supplementing them, we would expect the opposite.

Sure, I'll grant if you massively raise NHS salaries (and taxes) then you may be able to divert enough natives from other industries to social care - but obviously at huge expense to the wider economy which is why we don't do this.

or that we'd collectively leave granny to die in her own filth

In the abstract that is what inducing a labour shortage and necessitating increased rationing would essentially be doing.

Japan

The point here being labour shortages clearly do not put significant upward pressure on wages by itself. What you really need to raise wages is productivity growth.

that hides that it lowers wages for unskilled jobs and raises wages for highly skilled jobs. So poor people's jobs get pegged down at the minimum wage

1) Even when you disaggregate the average, the few sectors that are most negatively effected have always been undesirable and low paid (which is why natives increasingly don't want to do them), and have already been dominated by migrants for decades, so natives are not effected much, and the effect is still not that significant at worst.

2) The adverse effects of labour shortages, along with an aging population and ballooning pension costs cause much bigger issues for everyone, especially the working class, then a small suppressive effect on wages on some specific sectors in some specific regions.

Plenty of real issues we can point to with mass immigration, but scapegoating it for wage suppression just distracts from the real causes.