Well, first, I appreciate you calming down and having a rational discussion, as well as admitting you made a mistake, that's incredibly rare around here.
I actually have multiple problems with a minimum wage, but I'll start with the economic factors. The Government has no method of measuring supply and demand curves, nor do they have any skills which would allow any individual in Government to determine how businesses should be run, so not only are those in Government unqualified to determine this, but the Government itself, as an organization is completely incapable. As you noted in your post, increasing minimum wage doesn't actually help anyone, it just increases the cost of living, however that's not all there is to it. There are some jobs and some workers who are not worth the minimum wage, which actually causes a business to overpay some employees, and this causes them to be unable or unwilling to hire more employees.
Another problem with a minimum wage is that it prevents the lowest tier of jobs from teaching employees who become skilled enough from actually negotiating their wages, preventing them from learning such a skill at an early age. This is also in addition to the minimum wage gatekeeping newer businesses, preventing them from hiring employees that they can afford.
Not only all of this, but businesses are already naturally prevented from paying employees too low of a wage in the first place. Not only would they lose employees to competitors if they paid them too little, but they'd also be reducing the number of people in the area that can afford their products, even if people were willing to work for wages which were too low. Oh, and voluntary unions also can help people negotiate wages, though Government-Enforced Unions allow the union to bully the business, which increases the cost of living as well.
Some people also support Federal Aid for subverting the cost of living problem, however it actually pushes wages down, due to the businesses knowing that people can obtain it. One example is Walmart encouraging employees to apply for Food Stamps. Businesses do a similar thing when dealing with insurance; Knowing they're dealing with someone who has deeper pockets, they increase the cost of their services, they would otherwise ensure that consumers can afford their product, as they would otherwise make no money.
Now, with all of that explained, I also have an ethical problem with the minimum wage. It's the Government using force to tell businesses how much money they have to hand over to their employees, regardless of whether or not the position or laborer is actually worth that amount or not. This is the Government telling businesses, or really anyone who uses their property as a means of production, what they can and can't do with that property, forcing association.