A good overview of the case against minimum wages based on economic research & reasoning is this Cato Institute summary paper, "The Negative Effects of Minimum Wage Laws":

https://object.cato.org/pubs/pas/PA701.pdf

(This understanding was popular enough in 1987 that even the NYTimes endorsed it in an unsigned, editorial-board opinion piece – that meaning "this is the consensus opinion of the paper's editors": https://www.nytimes.com/1987/01/14/opinion/the-right-minimum... )

Fair enough, although the Cato report glosses over or ignores a lot of interesting subtleties in its cited sources (e.g. the FRB Atlanta report [15]). For arguments that minimum wages reduce employment, I'd sooner trust the various NBER working papers on the subject: https://www.nber.org/papers/w12663 https://www.nber.org/papers/w23532 https://www.nber.org/papers/w20724

Does any of that research suggest that a zero minimum wage is optimal in some sense? No. But I will grant you that raising the minimum wage might not do what we hope it would do.