There is clearly a counter-argument that the minimum wage is a market intervention and can cause inefficiencies (it may result in some folks who just can't be trained losing their $3 per hour jobs).

That's an inefficiency, but it seems to me that a far more central one is embedded in the assumptions of your toy model: how many unskilled jobs ($3) funge against skilled or semi-skilled ones ($10). In practice, it seems to me that the kind of jobs an employer can offer are often narrowly constrained by business requirements.

A factory owner, for example, might be able to retrain unskilled line workers (fitting Subwidget A to Subwidget B) to do semi-skilled work (operating a widget-fitting machine) for higher total productivity; that's consistent with your model's assumptions. But if you run, say, a hardware store, someone's got to stack shelves, mop the floors, and run the registers, all of which take roughly the same level of training, and there's only so many places you can squeeze out more per-body productivity by investing more. Anyone you have to fire because of minimum-wage laws there represents an economic loss: they aren't getting paid, and you aren't running as efficient a business as you could be.