All I know is in the UK it's not uncommon for jobs to get thousands of applications. I'm pretty confident the immigration is hitting the supply side more than demand. Most of this immigration is from low skilled workers on poverty wages, I'm struggling to see how this would massively increase demand elsewhere in the labour market.
Since immigration started increasing in the late 90's wages have been stagnant. Correlation doesn't equal causation, but hmm.
The UK outside London is IIRC poorer than all but one US state, and your housing policy makes it even harder to build new housing than California.
So you have much bigger problems. For there to be jobs there has to be industry first. That'd provide the demand.