It's not dirty pool at all. It's just a reductio ad absurdum argument, which is totally valid and rational.

You're missing the bigger point: the minimum wage law is what eliminates the jobs, by prohibiting them. Whether a business chooses to essentially rehire the same people at a higher wage is a different issue. It will be the case that some employees are not qualified to perform the jobs at the higher wage, and thus not profitable to employ at the higher wage. It is also the case that there is an inflection point where the job itself is not worth having a human perform at a certain wage level, which is the point of my reductio ad absurdum argument. That's why economists talk about affects "at the margin", which in this case are jobs that are only barely worth employing someone at the current wage, but would become not worth it if the wage or other costs were higher.

The study hardly rebuts the argument, since there are too many other factors involved besides minimum wage increases.