Hiring below minimum wage is illegal in most of the world.
'Illegal' doesn't mean 'bad'. Illegal labour is usually a straight-forward consequence of local or state law being unsuitable to market situation. Trying to ban working in certain conditions or for certain wage is like banning alcohol or drugs - you cannot wipe them all out, you just push it to the underground, make it out of control, untaxed and in some cases allow bad guys to earn profits from that.
And this law is especially important when you have a big pool of very cheap labor outside the borders. You are trying to imply that the situation is somehow in equilibrium, that any rise in wages due to need to hire legal workforce will be offset by higher product costs. That is a very myopic view. At best, maybe there is some kind of such global symmetry, when you consider both the natives and the foreigners, of all classes, and average it out.
A lot depends on what the current situation is and how you let the migration happen. If borders of your country are closed, you have a much poorer neighbour country and then suddenly the govt decides to get the borders wide open, a migration taking place as a result is unnaturally big and destroys stability of the local market. I agree migration to the country should be somehow regulated, but it should take place on the borders, not via economical regulations.
It seems you still look at only one aspect of the issue. Minimum wage gives some kind of protection to native workers against the immigrants. But it also harms the weakest, the least-qualified NATIVE people preventing them from taking their first legal job. If you are a poor 18-year-old kid from poor district of Detroit, finished some non-significant state school, have no experience at all, then nobody will hire you legally for a minimum wage as our work is simply not worth it. Either you get a badly-paid job now, but you work, learn, gain experience and after some time your wage may go up, or you are out of the labour market on the start. All those people living from social benefits, homeless on the streets of large cities, youngsters wandering around the blocks of flats with no perspectives - lots of them could find their place on the market if not state policy of regulating and taxing labour, including the a minimum wage.
What's worth saying here is that if you don't let cheap labour to the business in the country, then the business may leave the country and move to places with cheap labour. That's what basically took place in the last decades. Europe and America has been mostly deindustrialized and are based on services and trade now. Most big companies have their factories located in China, India, Vietnam etc. exactly because hiring people in Europe or USA is simply too expensive. So there you have your protection - nobody takes your jobs in the country, it's just the jobs that go elsewhere

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But I dont care about global view, I care primarily about the situation of people living inside my country, the local situation, that is who national government should serve. And the fact is, allowing big pool of very cheap unregulated labor onto domestic market forces native lower class to compete with them, and therefore harms their net economic situation, decreases their bargaining force. And any economic improvements of migrant workers due to having a job, and of natives due to having cheaper products, come at least partially at the expense of this domestic lower class. They lose more due to their wages being pushed down and increase in unemployment than they would lose due to slightly more costly food.
In a truly free global market with zero trade, migration and work barriers, local inequality will inevitably reflect global one. You will have significant number of people living in absolute poverty inside your country. And your natives will have to compete with them. That is not acceptable to me, and certain amount of protectionist policies is justified in preventing that.
When Poland entered the EU, about 2 milion Poles left the country seeking a better life in western Europe, mostly in Germany, UK and Ireland. Most of them found it, as they are hard-working and reliable, often highly-qualified, taking jobs the native British were unager to take. British employers, leaking enough labour, were eager to hire them. Both sides benefit with such a deal. If I were a Polish migrant in a UK, worked long hours, paid my taxes, my employer was satisfied with me and wanted to hire me further, then what right can the state have to intervene in our voluntary deal?