were I running a business, I wouldn't personally offer unpaid internships

"Hi, Mr. b_b, I am not old enough to legally work, but I would love to pick up balls on your golf course and clean carts and just hang around and learn about the golf business because I dream of owning my own course one day. Also, someone told me that 'The ability to make decisions for oneself is among the most important things that makes us human' and I decided that staying home and watching TV is a waste of time."

"Scram, kid."

The evidence that I read seems to suggest that offering a low-to-fair wage for these jobs actually attracts higher quality, more diverse groups of candidates than unpaid positions

Of course offering higher wages attracts higher quality workers. That's the whole point. You get what you pay for, and better-qualified workers will compete for richer compensation, pushing others out. I thought you were worried about "less well off workers"?

"The chief practical negative of minimum wage is increased unemployment among the most vulnerable workers, who would most benefit from greater access to employment." The numbers have improved since the last time I checked, but the pattern is still the same:

The number of unemployed youth was 2.8 million in July 2015, down from 3.4 million a year earlier. The youth unemployment rate was 12.2 percent in July 2015, 2.1 percentage points less than a year before. Among the major demographic groups, July unemployment rates were lower than the prior year for young men (12.7 percent), women (11.7 percent), whites (10.3 percent), blacks (20.7 percent), and Hispanics (12.7 percent). The youth jobless rate changed little for Asians (10.7 percent).

it appears that taking an unpaid internship often doesn't increase one's job prospects

Compared to what? If you are saying one's future job prospects would increase more if one got a paid position instead of taking an internship, I think that's reasonable.

If you are saying one's future job prospects would increase more if one remained unemployed, I doubt it.

If an unemployed person has to choose between an unpaid internship and a similar but paid position, I don't think we need FLSA to nudge them in the right direction.