Card doesn't rely on irrational firms. He showed with data that labor demand is inelastic at minimum wage in the region studied. When this is true, the benefits of increasing minimum wage outweigh the costs.
So then they republished and it was like "well if we only look at this group and we take out these and adjust this number" they were able to recover their results but the data hacking was pretty terrible.
If you ate going to write a study like that, relying on phone data screams bias (and these were two economists that couldn't claim they didn't know better).