Having people who have been living in the country for decades, but have the threat of leaving anytime they get fired from job or due to visa issues doesn't encourage them to "integrate into the society they are joining".

If anything, the way the country caps work in the US right now make integration harder, because no matter how much they try to integrate and be part of the local community, they could be kicked out at any time. That just encourages people to have one foot out the door at all times.