If the minimum wage is below a living wage, it is artificially low. The market does not pay you more just because you need more to survive, hence the need for a solid wage floor. The thesis, high level, is: do we care about maximizing for profits (no or artificially low minimum wage)? Or some less profit and humans being able to live with dignity and some form of psychological safety (minimum wage aligned to living wage).

Minimum wages increase the cost of living by making low level services cost more. This is a circular problem.

Also the definition of "living wage" varies wildly depending on what kitchen sink people want to throw into it. People seem to think that everyone deserves to make enough money for luxuries whereas that shouldn't be the case in general.