But if you want to talk about employees, what you're saying isn't really true.
Sourcing, recruiting, interviewing, hiring, and relocating immigrant workers is very expensive. For an immigrant to outcompete a citizen on cost they have to be significantly cheaper than the citizen - but if they're on an H1B you have to pay them the prevailing wage for their position! So there's a floor on just how much immigrant workers can undercut citizen workers. Immigration certainly puts downward pressure on wages across a whole industry, but looking at software in particular we see wages having risen consistently for decades now. You can look at what companies are paying their H1B software workers and see that it's typically a very generous wage by US standards.
But, weirdly, a majority of US CS graduates don't work in software. It doesn't make sense that they would be holding out for more money than immigrant workers to the point that they end up exiting the industry altogether and taking a less lucrative job. Not to me, at least.
I think what you're saying, if it's happening at all, is likely only happening at the bottom end of the market. Which kind of proves my point.