The only shortage is in pay. I and many other people left trades becauses wages were crap for jobs that take 20 years of your life away. Despite the memes about trades, the majority of trade workers are not making bank unless they have an unusual specialty and a union. 1 plumber in Manhatten doing making bank doesn't make the plumbers all across the rest of the nation wealthy.

Why can't we train our own citizens to become construction workers?

If you claim that people "don't want to work" for those wages, then isn't immigration keeping our wages down?

Wouldn't business owners pay more for labor if there wasn't a stead supply of under the table illegal immigrants doing it for less?

If there is reduced demand for new housing due to increased inventory, there's less of a demand on construction for new housing. The expenses involved in new builds is both labor and building codes/restrictions - construction firms undercut Americans by hiring foreign labor that they can pay under the table.

I would much prefer Americans pay a premium for skilled American labor and choose to live in more modestly-sized homes.

We also need immigration because it grows the GDP, makes for a dynamic and resilient economy, increases scientific and technological output and raises the standard of living.

Immigration is what made America great.