The only way you can claim that their argument is that is if you believe that the minimum wage simply means "paying poor people more", and nothing more, which is an absurd simplification, and ignores a lot of nuance (ex. employers don't have to pay the minimum wage, they can simply fire the employee [0]).

If it's really the case that minimum wage means paying poor people more, surely we could just set the minimum wage at $30/hr and pay poor people even more.

I'm not even opposed to the minimum wage myself. I just don't think it's good to simplify a complex situation, and then use that to strawman someone's legitimate point (that it is not a given that minimum wages will reduces poverty rate).

[0] - Another example is that minimum wages vary across jurisdictions, and employers are able to arbitrage that fairly effectively for many jobs.

Minimum-wage employees tend not to be people that can be easily eliminated or offshored. You really haven't thought any of this through, have you?