You are wrong. The idea would be sound if there was equal negotiation power. But there is not. There are all sorts of rules that prevent labor from fairly negotiating with management. Managements ability to manipulate the labor pool, the basic cost of living, and successfully petition government for assistance when they are financially hurt by labor, constitutes a much larger redistribution program than minimum wage.
Abolish government corporate subsidies, institute a tariff on all imported goods, and remove all the varied union breaking rules, we could consider removing the minimum wage. But of course if you did that wages would be much higher than the minimum wage anyway.
We will know when the negotiation power is equalized when corporations makes no more profit than their employees.
Here is how you fairly negotiate: Observe what the market is paying other workers with skills similar to yours. Ask for that wage, and if it is given, begin working. If you don't like the wage, don't take the job. If no available job pays you what you want, then you need to make yourself better.
You are not worth what you want to be paid or what you need to be paid. You are worth what your level of talent and experience are getting paid to other similar workers.
It's totally fair. You don't have a right to coerce someone to pay you more than they want to. That is called slavery.
As far as abolishing government subsidies, yes, I agree with that 100%. As far as tariffs? No go. Tariffs stifle competition and reward inefficiency and incompetence. I don't care where something is made, I only care how well it is made. I don't need my fellow citizens thwarting trade to protect domestic mediocrity.
As far as union breaking rules, I am in favor of no union rules at all. Unions should not be allowed to coerce anyone into joining or paying dues. If a union can convince a bunch of workers to join for purposes of collective bargaining on a purely voluntary basis, I am good with that. But NO LEGAL PROTECTIONS. No mandatory dues. Either it is voluntary and free or I don't want it.
The minimum wage is coercive labor cost price fixing backed by the police power of the state. That's wrong.
If someone wants higher wages, they need to make themselves a better person. No one deserves to be paid well to do a common or menial task that anyone can do. Common skills don't pay well, so you have to make yourself a little bit more uncommon to expect high pay. That is fair, logical, moral, rational, and just.