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We have labor laws in the US because 'some people will take the deal.'"
No, we have labor laws to subsidize labor unions by driving up the cost of labor. It's one of the reasons we have rust belts instead of factory belts.
"You are flooding the market with artificially cheap labor, making it impossible for others who can't work for free to compete."
If a person's labor has lower value than free, they have a problem. That problem is not competition.
"Worse, as a business owner, how am I to compete with someone getting free labor while I'm paying my employees like a sucker?"
Logical thinking fail. "Artificially cheap labor", to use your words, drives down your costs for both labor and supplies.
In any event, unpaid internships are consistent with the principle of free association: the participants do not harm each other, and do not coerce anyone else. Restricting unpaid internships is morally equivalent to banning a church that does not charge for admission, on the theory that they unfairly compete with movie theaters.