There's something more insidious about this than "cheap labor", because "big companies" are not the primary employers of illegals.
It's the landscapers, the butchers, the contractors, and small/medium businesses that are hiring illegals. They don't really have a lobby in DC, and they aren't supported by the unions.
If I drive down to Culmore Shopping Center in Falls Church Virginia today, there won't be any "big business" supporting those 50 to 125 guys milling in the 7-11 parking lot.
Similarly, there are lots of medium/small businesses hiring legal immigrants who are nonetheless "unskilled" by American standards and don't speak English, but they at least get minimum wage.
So where is the pressure for illegals? It's not "big business", it's obviously not the unions, so.... only PC politicians???
Artificial wage suppression lowers pay scales at all levels. And that's good for big business.
Corporate hotels are a big supporter of relaxed immigration rules along with corporate farms and construction.