Several things come to mind, here's a few.

In global capitalism the owning class has access to this cheap labor already, it's just out of sight in the West. The system prefers this because this justifies the low wages and can be used to lower wages in the core as well. This hides the oppression from the Western consumers and keeps their consumer power high enough to buy all the treats that creates the profit for the capitalist.

So I think that the walled garden of consumers and higher prices is needed to uphold the divide of consumer status and collection of profit.

There are also several entire industries in the service sectors in the core that manage and control the labor surplus. Which creates bs job type finance movement and policing etc. Not to mention how borders and border control create this "profit from nothing". The service sectors of post industrial countries are huge and they all rely on these divides to justify their existence.

If migration was free it would also reveal to the masses the pointlesness of national borders. It would bright forward a call for fair wages on a global scale. If they could no longer outsource labor, but the labor would come to them, there would have to be higher wages or there would likely be revolution.

There are also many unemployed people in imperial core countries that at the same time do brain drain colonialism from the global south. There isn't a lack of laborers, but a lack of "skilled labor" in the service sectors because there is no industry. The industry is in China and elsewhere now where there are plenty of people to do the work. We just have the service sector stuff now and some low wage jobs that nobody can be forced to do, not unless oppressed under the migration apparatus or the status of modern slavery of being paperless in one of these countries.

I also think that since access to the core is presented like some coveted achievement that people are forced to fight for, it creates a system that just re-enforces itself.

And people with no papers or status can be exploited uncheched inside the core as well and this too requires the anti-migrant sentiment and controls to exist. If everybody had a right to be there, those everyone would be a lot harder to exploit.

For immigration, market shocks of cheap labor would hurt incumbents in the short term. Most well established companies already pay a premium for labor so cheap labor would only serve to increase competition, as it will take them time to replace their current staff with cheaper labor. The middle class also doesn't want their pay to decrease, but fail to see that would be generally better off. So most capital wants status-quo, which is a strict immigration policy with slow growth. Deportations seems more like a social filter over a particular demographic, we don't see highly skilled engineers get deported for example.

All great answers. But there’s something people are missing, and it’s a trap created by modern neo-liberal economic thinking.

Humans are not perfectly rational entities. Capital is not an external alien force that is perfectly rational either (yet). It moves and is brought to life by human hands.

The ruling forces at the moment are losing their grip. They are desperate. Somethings ARE running away from them.

They are acting irrationally all the time. They are shooting themselves on the foot, over and over. This is one of those things imo.