>> Well, the simple fact is, empirical studies of minimum wages just don't show their having a macroeconomic effect on employment that overwhelms other effects
You don't get it. There's no need for "empirical studies", because it's a simple matter of reasoning.
Suppose Person A is fresh out of high school or something, and Company B might be interested in hiring him as some kind of entry level worker. Person A has basically no experience in anything, so he's worth very little to the employer. He really wants a job though, because he needs to start some kind of career to get by in life.
Suppose Company B consider's Person A's meager productivity worth $6 per hour, but the minimum wage is set to $10 per hour. Will Person A get the job?
Well, no. The minimum wage prevents Person A from getting a job, and thus, getting experience and advancing his skills and future salary potential. Everyone loses. Person A has no job and no further experience, and Company B has no employee and a task left undone.
If you still don't get it, imagine the minimum wage being set at $1000 per hour. If it's all good, then the higher you set it, the MOAR good will come out of it, right?