Yes! I think that brilliant statement of yours should be the GOP campaign 2020 slogan and it's main meme:
"If someone else is willing to do a task cheaper than you it’s not the government’s problem to solve."
I am sure the GOP can take all 50 states with a campaign slogan like that.
Such a statement has NOTHING to do with illegal immigration. It has EVERYTHING to do with allowing competition in the marketplace being the final arbiter of cost/product pricing rather than some government dictated outcome. All Americans would benefit from being able to buy products cheaper and being able to compete for work that others currently are charging too much for. That's how a free market works. The GOP could certainly say they will remove government interference in the free market and get tons of support.