Minimum wage isn't just flat for a decade, it's not being adjusted for inflation. This means minimum wage workers have been getting a pay cut over time. The debate isn't just whether we should raise minimum wage, it's whether we should be cutting it. By allowing it to remain stagnant we're making the default decision to cut.
I don't understand why this isn't a red herring. Minimum wage may be remaining stagnant but total employee compensation has been rising steadily, it's just that most of that compensation is in the form of spending on employee benefits like insurance. If people quit expecting employers to provide them with anything except for cold hard cash then you'd likely see wages increasing with time. To expand a bit more on this…
If salaries are going down due to inflation, then you may have a point but it's a perverse one; medical costs are going up way faster than anything else.
Both are reasons to support socialized medical coverage, but they don't really enter into this particular line of discussion.