Bryan's central point: Economic theory lets you vastly broaden the range of experience that you can bring to one question -- the effect of minimum wages in Seattle, for example. Economic theory also forces logical consistency that would not otherwise be obvious. You can't argue that the labor demand curve is vertical today, for the minimum wage, and horizontal tomorrow, for immigrants. There is one labor demand curve, and it is what it is. Economics lets one experience illuminate the other, and done right forces politically uncomfortable consistency on those views. You can't argue that sticky too-high wages cause unemployment in recessions and in Greece, and not argue that sticky too-high wages from minimum wages laws do not cause unemployment in Seattle.
This kind of integrated thinking is far too rare in evaluating economic policies. But that's the fault of economists, not of economics.
Research doesn't have to officially be about the minimum wage to be highly relevant to the debate. All of the following empirical literatures support the orthodox view that the minimum wage has pronounced disemployment effects:
1. The literature on the effect of low-skilled immigration on native wages. A strong consensus finds that large increases in low-skilled immigration have little effect on low-skilled native wages. David Card himself is a major contributor here, most famously for his study of the Mariel boatlift. These results imply a highly elastic demand curve for low-skilled labor, which in turn implies a large disemployment effect of the minimum wage.
This consensus among immigration researchers is so strong that George Borjas titled his dissenting paper "The Labor Demand Curve Is Downward Sloping." If this were a paper on the minimum wage, readers would assume Borjas was arguing that the labor demand curve is downward-sloping rather than vertical. Since he's writing about immigration, however, he's actually claiming the labor demand curve is downward-sloping rather than horizontal!
2. The literature on the effect of European labor market regulation. Most economists who study European labor markets admit that strict labor market regulations are an important cause of high long-term unemployment. When I ask random European economists, they tell me, "The economics is clear; the problem is politics," meaning that European governments are afraid to embrace the deregulation they know they need to restore full employment. To be fair, high minimum wages are only one facet of European labor market regulation. But if you find that one kind of regulation that raises labor costs reduces employment, the reasonable inference to draw is that any regulation that raises labor costs has similar effects - including, of course, the minimum wage.
3. The literature on the effects of price controls in general. There are vast empirical literatures studying the effects of price controls of housing (rent control), agriculture (price supports), energy (oil and gas price controls), banking (Regulation Q) etc. Each of these literatures bolsters the textbook story about the effect of price controls - and therefore ipso facto bolsters the textbook story about the effect of price controls in the labor market.
If you object, "Evidence on rent control is only relevant for housing markets, not labor markets," I'll retort, "In that case, evidence on the minimum wage in New Jersey and Pennsylvania in the 1990s is only relevant for those two states during that decade." My point: If you can't generalize empirical results from one market to another, you can't generalize empirical results from one state to another, or one era to another. And if that's what you think, empirical work is a waste of time.
4. The literature on Keynesian macroeconomics. If you're even mildly Keynesian, you know that downward nominal wage rigidity occasionally leads to lots of involuntary unemployment. If, like most Keynesians, you think that your view is backed by overwhelming empirical evidence, I have a challenge for you: Explain why market-driven downward nominal wage rigidity leads to unemployment without implying that a government-imposed minimum wage leads to unemployment. The challenge is tough because the whole point of the minimum wage is to intensify what Keynesians correctly see as the fundamental cause of unemployment: The failure of nominal wages to fall until the market clears.