There are only 2 million unemployed Americans right now. Most of the illegal immigrants have jobs and fill in the gaps, such as working on farms and factories. If the 20 million illegal immigrants are deported, want that create a massive laborer shortage? Won’t the work follow the workers to Mexico?
I wouldn’t say there’s no such thing as a labor shortage, we just don’t really have one now and haven’t for a while. It’s like saying there’s no such thing as a food shortage because in periods of high demand you can always just pay exponentially more money and get it.
If the rate at which labor costs are rising far outstrips the rate at which demand is growing across an industry, not just a business, that’s a sign that the supply of workers is lagging behind the demand growth. Usually seen when there’s a time lag between when demand can start to rise and people from other sectors can move over, like in medical fields or fields with high technical requirements.
It’s still not the workers responsibility to take lower wages to keep a business afloat, but that doesn’t mean there haven’t been times when it’s been legitimately infeasible to fill a position.
Businesses and in some cases governments just need to be forward thinking and give incentive to start training for the career before demand starts to outstrip supply.
Smart places with nurse shortages will do stuff like pay for your training in exchange for a set number of years working for them at a market wage.