Yup. So having minimum wages to ensure people can make a living wage is exactly the same as having to have price fixing for food to ensure people can eat. It's a knee jerk response to a fundamental problem (that workers make too little in relation to price levels).

The problem with the US discourse on the subject is that it's a debate for/against higher minimum wages rather than a debate of "what regulations need to change so that the minimum wage without regulation is a livable wage".