Something I advocate for in the UK is that we should actively suppress wages of industries like healthcare and agricultural with cheap foreign labour, while providing access to good and affordable education so that native workers maximally benefit from the migrant labour underclass. Specifically I think we should bring foreign workers here on mass to work in specific industries then deport them if they're unable to find work. We should ensure they're not allowed to work in other industries to reduce labour competition with natives working in sectors for which we don't want to suppress wages.
That said, some people disagree with the ethics of importing a migrant underclass which we're knowingly underpaying, but I'd agree with you that the current system benefits us, although it could be better with more immigration and labour controls.