The problem with the letter is that he buys into the notion that empirical studies can somehow prove or disprove logical deductions. He attacks the empirical studies of those who claim minimum wages do not decrease employment by listing other empirical studies that point in a different direction. He does this rather than recognize the fact that you can't conduct empirical studies in the science of economics, in the first place, the way you can in the physical sciences. He is getting sucked into the trap.
The simple fact is that if you force people to pay more for something, they will buy less of it. No empirical tests, required. It's basic logic.
This is a basic supply and demand curve, graphically making the point.
Here's a graph showing the shortages and over-supplies that would occur at prices different from the market price. The market price being where supply and demand are equal. A minimum wage increase above the market price would occur where instead of a market clearing wage of p*, the wage would be above p*. At all minimum wage levels above p*, the quantity of workers hired would fall.
There are no empirical studies that can refute this. It is pure logic. And no empirical studies are needed to prove the argument. They can't.
Anyone using empirical data to try and prove or disprove logic is a quack.
Although it appears he wasn't familiar with the proper (deductive) methodology for the sciences of human action, developed by the Austrian school, Richard Feynman understood at a gut level that the empirical testing being done in the social sciences is hocus pocus.