If someone isn't worth the current minimnum wage, then they should be fired. If someone's work ethic is so slack that they are pissing their time on a minimum wage job, the company should fire them and hire someone else, rather than make a claim that they should pay a lazy person less.
The lowest salary I pay (I own a company) is $15.00 an hour, which is twice what the proposed minimum wage is. 15 an hour is a living wage in my opinion. Not a great living wage, but a living one. You're right that there aren't many people on the minimum wage currently, and those that are, are generally transitory workers heading up to a higher wage, or part time work. The problem is that it's not all, and it needs to be reflected.
The minimum wage is about making sure that people aren't exploited. It's one of the reasons that our major cities are surrounded with places like this...
Holding the minimum wage down, or even getting rid of it as some conservatives advocate, would inevitably create these situations. Wages would remain depressed with no reason to rise, while the cost of living, of rent, of land, of food, would all increase. It wouldn't happen overnight, and might take years or even decades, but it would happen. The minimum wage is meant to protect the workers, to assure than every man and woman who works a full time job, get paid an amount which allows them to live in a reasonable manner. THat picture does not represent a reasonable manner.
Plus the minimum wage affects more than just those that make the minimum, but those above it. Someone making 7 an hour now, will see their pay increased above the minimum wage as well. Someone making 8 might now make 10, and so on. It helps more than just those on the lowest rung. In addition more money in the pockets of the people who will spend it has time and again been shown to boost the economy. People who make 10 dollars an hour spend it all, and each of their dollars circulates in the economy more. Businesses get that money as spending increases and the econonmy increases around it.
So I disagree with your reasoning. It will help many people, those few that stay on minimum wage for a long term, and the large number of people whos wages hover within 5-10 dollars an hour of that wage. All of them will benefit. If someone has a poor work ethic it means they should be fired, not that they shouldn't earn a decent wage.