Henryk
Well, you do have some knowledge of Canada's "Foreign Temporary Worker" program. I'll just call it FTW.
When an employer has trouble finding workers, they can make arrangements with the federal government to fill spots with FTW workers. The company has to first prove that Canadians don't want these jobs. The workers are not allowed to quit and work elsewhere. When their work is terminated, they go back home.
Quite a few of these FTWs do stay with the original employer--and this leads to permanent residency and Canadian citizenship. But it still means playing by the rules.
As far as "needs of the labor market" is concerned, this is a bit of a moot point. The companies hiring FTWs usually aren't paying "market rates" for wages to attract Canadians, who have many more options for employment. A good example is our agricultural industry where labor is still a big component for harvesting crops. A FTW is happy with $10 an hour for picking apples; it might take $30 an hour to attract Canadians into this temporary occupation. We could have a long discussion whether this is good for the economy or labor market, but this is essentially how the Canadian system works.
A FTW really doesn't have much choice. If he quits and goes on the black market for his labor, he might stay undercover for 5 years before being found out and sent home. Employers who pay employers "under the table" face lots of fines, whether their workers are FTW or regular Canadians. It hard for workers in Canada to "work for cash" on a regular and sustainable basis.
In the USA, an illegal worker can hop from job to job for decades and not be discovered. Or sometimes he/she stays in the same place for decades and not be discovered. Many employers know they are directly or indirectly hiring illegal immigrants and also know they won't be punished for it. That is a big difference that makes USA more attractive for illegal immigration. When they know relatives or neighbors in the USA working under the table with no consequences, they are more likely to go there and get similar employment.
The elite benefit from such a system. They can hire cheaper people with little regulation. The working Americans suffer because their job market has been distorted against their favor.