My rebuttal is that no one is arguing to encourage illegal labor and immigration.

"The US economy cannot possibly sustain the type of deportations that have been promised" is not saying "an economy cannot function without illegal labor." It is saying exactly what it says: an economy cannot sustain (i.e. remain healthy through) the mass expulsion of a huge portion of its lowest level labor force.

I made it explicitly clear that I am talking about an (almost certainly) non-permanent problem: "I also believe America will almost certainly recover from whatever dark period it's (probably) about to endure."

By analogy: The statement that the US economy cannot sustain a 90% reduction in equity values market-wide doesn't mean an economy can't exist that's 10% the size of the United States'. It doesn't mean an economy 10% of the size of the United States' can't grow to become as big or bigger than the United States'. It doesn't mean a 90% drop in equity values would delete the United States from existence.

It means that a sudden 90% drop in equity values would shock the system in intensely undesirable ways.

Mass deportations as proposed would be a gigantic shock to the system, and that shock will almost certainly make the US an undesirable trading partner for some time.