What harm do minimum wages do?

https://universalbasicservices.org/

Wages are a cost to a company. There are other costs to a company. Companies also return a profit to the owners. If a company's costs are too high, they my not return a profit; they may not be able to pay their creditors.

My dad ran his own very small business. I am familiar with a business not making a profit for a period of time. I am familiar with workers who are not profitable to employ.

It is a reduction to absurdity to state that minimum wage "makes it illegal to work unless you can produce $X value per hour."

There are interesting, non-absurd discussions about the effects of minimum wage, but this is not one of them.

The marginal value of what he produces. Which means that if there is a minimum wage in effect, the employer might be forced to pay the worker more than the marginal value of what he produces. Which in practice means that employer starts looking for ways to eliminate that job, since it costs more than it is worth. If employers do not do that in the presence of minimum wage laws, it simply means the laws are having zero effect because the minimum wage is below the free market equilibrium wage for that job anyway.

> The labor market is not a free market

It is if there are no laws fixing prices, i.e., no minimum wage laws.

None of the things you list prevent a market from being a free market. A free market is not a market where everyone has equal negotiating power, equal information, or equal ability to deal with some good or service not being available. A free market is a market in which all transactions are voluntary--i.e., a transaction happens only if both parties believe that that transaction, at that price, will make them better off. Any third party dictating the terms of transactions--of which minimum wage laws are just one of many examples--makes the market a non-free market. The fact that the third party believes that it is "improving" something doesn't change the economic fact that it is forcing the market to not be free, and therefore making the market less efficient.

> Companies will always push for lower wages wherever they can, without a care for the actual wealth created by the worker

In a free market, they can't, because their competitors will be willing to pay more for workers that are worth more in terms of productivity. Tech companies are good examples: they are willing to pay a high premium for workers that can create a lot of value by writing good code and more generally doing good, reliable software engineering. Companies that try to pay less for such workers simply don't get them.

> The only way to counter that push is by giving more power to the worker

No, that doesn't counter anything, because "the worker" is not one person and does not have one interest. The best way to help workers is to raise their productivity, so that the work they do is worth more and can command higher wages. But really, even that misstates it, because it treats the workers as though they weren't capable of raising their own productivity if given the opportunity, and need "help" in doing so. Which is the big problem with all the grand schemes to "help the worker": it's people with no skin in the game themselves trying to dictate to everyone else how they should play.

It's not like companies want to pay for shipping of products from halfway across the world, but because of minimum wage it makes it more profitable to do so. Which results in jobs leaving to other countries.

Without minimum, those workers wages would have simply gone down over time, allowing them some time to find other opportunities. Instead of just one day the plants up and decides to move to Asia and you're out of a job.