There's nothing 'magic' about cutting migration, it is quite easy for a government to do if they want.
Migration never fixes general skills shortages (though it can be useful to address very specific shortages) because - shock, horror - migrants themselves have needs, too. or did you think that migrants won't require teachers to teach their children, or nurses to tend to them when they are sick, or builders to build their houses, or roads to drive on. This is why the population ponzi never ends, it always demands MOAR PEOPLE.
