Anubis, you are spot on with your cause and effect relationship regarding wages and the demand for labor. As someone who has studied economics (doubt DP has), we know that by raising minimum wages and the cost of labor on one particular cost curve, you cause all the other cost curves to shift in response. In other words, you create both a substitution effect for laborers who were earning a premium to minimum wage who now find themselves at minimum wage as well as affecting the marginal supplier of labor on the supply curve. This is all basic math and not opinion. And you nailed it.
As was pointed out, if you have two groups of people, one currently at 7.25 an hour and let's say another group at 15 to keep things simple. Say the group at 15 an hour has some discernible skill that has afforded them this wage in the marketplace. They possibly have a more stressful job, have greater responsibility and some skill they had to attain. Now you shift the labor curve so that those with no skills, no responsibility and no stress are now making the same wages as the 15 group. Well, this is where the substitution effect kicks in. Those in the 15 group are now sitting on the wrong indifference curve. They are making the same wages as the min group but with harder jobs that require more skill and more responsibility. The optimal strategy for them is to "shift" to the lower curve. That is, they can trade in their current job for the lesser job and earn the same wage. They now have increased their personal utility at no cost to them. But the rub is, they just now took the jobs meant for those in the 7.25 min group. Which means they lose their jobs. Employers will jump at the chance to higher a higher quality worker at the same rate they would have to pay the old 7.25 min group. So the firm has increased their utility as well given the new wage floor. Each actor in this market has maximized their utility while the low end group finds themselves priced out of the labor pool.
This is why economists have never favored raising the min wage. And most of them are center left politically. Instead they have favored for the increase in welfare and gov't benefits. And yes, your observation and one that is frequently brought up is, if more is better, then why stop at 15 an hour? Why not raise min wage to 100,000 a year. Obviously more is not better. So the next question is, is the labor market operating inefficiently. In other words, is the floor for labor too low relative to the value and productivity they provide. Now this is a genuine argument. You are not going to find widespread agreement here. My opinion is that low end of the labor market offers little to no value to most firms. If firms can afford to enter the labor market and purchase this labor at this rate, they are doing so because they are earning a marginal return on that labor in that particular moment in time that justifies the expense.
On the other side, the marginal supplier of labor is highly sensitive to his marginal cost. As we know from economics, the marginal supplier of labor is the last supplier of labor on the supply curve with the smallest margins. Hence why he is "operating on the margin". Any slight increase in cost and he falls off the supply curve. This means he can no longer be a buyer of labor at this higher cost level. This means all the people working for him are out of a job. Now not every firm is operating on the margin. But enough firms are and they will reduce the amount of labor in the marketplace. This is not a controversial subjected and it's implications are widely accepted across economists all political persuasions. This exercise has been researched and researched to death. This example is used in EVERY micro economics text book in the country. Every 10 years though it gets brought up again by someone new thinking they have come across some new idea that has not been well researched.